1 
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1  III  I     nifiii 

:|i  111        liiilll  III 


MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 
Octavo,  Cloth,  $1.00  each. 

No.  1.     HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 

No.  2.    SYNTHETIC  PROJECTIVE   GEOMETRY. 

By  George  Uruce  Halsted. 

No.  3.    DETERMINANTS. 

By  Laenas  Giffokd  Weld. 

No.  4.    HYPERBOLIC    FUNCTIONS. 

By  James  McMahon. 

No.  5.     HARMONIC   FUNCTIONS. 

By  William  E.  Byerly. 

No.  6.    QR\SSMANN'S  SPACE   ANALYSIS. 

By  Edward  W.   Hyde. 

No.  7.     PROBABILITY    AND  THEORY    OF    ERRORS. 

By  Robert  S.  Woodward. 

No.  8.    VECTOR  ANALYSIS   AND  QUATERNIONS. 

By  Alexander  Macfarlanr. 

No.  9      DIFFERENTIAL   EQUATIONS. 

By  Willia:\i  Woolsey  Johnson. 

No.  10     THE  SOLUTION  OF   EQUATIONS. 

By  Mansfield  Mekrim.\n. 

No.  11.    FUNCTIONS  OF  A   COMPLEX  VARIABLE. 

By  Thomas  S.  Fiskb. 

PUr.T.ISHED    BY 

JOHN   WILEY  &   SONS,   NEW  YORK. 
CHAPMAN  &  HALL,  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  6. 

GRASSMANN'S 
SPACE    ANALYSIS 


BY 


EDWARD   W.    HYDE, 

ACTl'AKY   OF    THE   COLUMBIA    INSURANCE    COMPANY. 


FOURTH    EDITION. 

FIRST    THOUSAND. 


NEW  YORK: 

JOHN    WILEY    &    SONS. 

London:    CHAPMAN  &   HALL,    Limited. 

1906. 


Copyright,  1896, 

BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD 

UNDER   THE    TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,   January,  1906. 


ROBKRT  DRUMMOND,    PRINTER,    NEW   YORK. 


EDITORS'   PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theoiy  of  num. 
bers,  the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubHcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


236367 


AUTHOR'S   PREPACK. 


This  little  book  is  an  attempt  to  present  simply  and  con- 
cisely the  elemental-}'  principles  of  the  "Extensive  Analysis'"  as 
fully  developed  in  the  comprehensive  treatises  of  Hermann 
Grassmann,  restricting  the  treatment  however  to  the  geometry 
of  two  and  three  dimensional  space.  It  is  designed  to  set  forth, 
as  far  as  is  possible  in  so  brief  a  work,  the  remarkable  adapta- 
bility and  effectiveness  of  the  methods  used  as  applied  to  the 
various  problems  and  operations  of  geometry  and  mechanics. 

The  ideas  of  direction  and  position  appear  to  the  writer  to 
be  as  simple  and  fundamental  as  that  of  magnitude,  and  an 
algebra  which  deals  directly  with  all  three  of  these  ideas  should 
not  be  greatly  more  difficult  than  the  ordinary  one,  which  deals 
with  magnitude  only.  The  result  of  using  such  an  algebra  is  an 
extraordinary  gain  in  the  brevity  of  operations  and  the  expres- 
siveness of  formulas  and  equations. 

Some  of  the  terms  belonging  to  this  general  subject  are  fre- 
quently employed  in  modern  text-books  on  mechanics  and  physics, 
even  when  no  use  is  made  of  the  algebraic  systems  from  which 
they  are  derived. 

It  is  hoped  that  this  little  book  may  do  something  to  interest 
students,  and  to  help  toward  bringing  in  the  time  when  the 
methods  as  well  as  the  ideas  of  this  calculus  shall  come  into 
general  use. 

Cincinnati,  O.,  December,  1905. 


CONTENTS. 


Art.     I.  Explanations  and  Definitions Page  8 

2.  Sum  and  Difference  of  Two  Points 9 

3.  Sum  of  Two  Weighted  Points 12 

4.  Sum  of  any  Number  of  Points 15 

5.  Reference  Systems 20 

6.  Nature  of  Goemetric  Multiplication 24 

7.  Planimetric  Products 26 

8.  The  Complement 33 

9.  Equations  of  Condition  and  Formulas 39 

10.  Stereometric  Products 44 

11.  The  Complement  in  Solid  Space 50 

12.  Addition  of  Sects  in  Solid  Space 53 


GRASSMANN'S  SPACE   ANALYSIS. 


Introduction. 

The  title  adopted  for  this  brief  and  elementary  discussion  of 
Grassmann's  methods  indicates  at  once  its  Hmitations;  for  his 
theory  in  its  fullness  treats  of  the  linear  relation 

p  =  lxkek,     (k  =  o,  1,  2,  .  .  .fi), 

in  which  n  may  be  any  positive  whole  number,  Xq,  Xi  .  .  .  any 
numbers  whatever,  and  Cq,  ei  .  .  .  units  of  any  kind  which  are 
susceptible  of  being  related  to  each  other  by  such  an  equation 
as  the  above.  Our  treatment  extends  only  to  the  case  when  n 
does  not  exceed  three,  and  Cq,  e^  .  .  .  are  points,  or  point  products. 

Grassmann's  tirst  pubHcation  of  his  new  system  was  made 
in  1844  in  a  book  entitled  "Die  Lineale  Ausdehnungslehre  Ein 
Neuer  Zweig  der  Mathematik."  His  novel  and  fruitful  ideas 
were  however  presented  in  a  somewhat  abstruse  and  unusual 
form,  with  the  result,  as  the  author  himself  states  in  the  preface 
to  the  second  edition  issued  in  1878,  that  scarcely  any  notice 
was  taken  of  the  book  by  Mathematicians. 

He  was  finally  comdnced  that  it  would  be  necessary  to  treat 
the  subject  in  an  entirely  dift'erent  manner  in  order  to  gain  the 
attention  of  the  mathematical  world.  Accordingly  in  1862  he 
pubhshed  "Die  Ausdehnungslehre,  vollstandig  und  in  strenger 
Form  bearbeitet,"  in  which  the  treatment  is  algebraic,  and  is 
developed  from  the  equation  given  above. 

Since  that  time  his  great  work  has  been  more  fully  appreciated, 
but  not  even  yet,  in  the  opinion  of  the  writer,  at  its  real  value.. 


:S.  crassmann's  space  analysis. 

Hamilton  first  gained  the  ear  of  the  English-speaking  world 
for  his  Quaternion  methods,  and  was  fortunate  in  having  some 
very  zealous  adherents  and  interpreters  who  made  propaganda 
for  them,  and  were  inclined  to  undervalue  work  not  originating 
in  England. 

It  is  hoped  that  the  following  brief  presentation  of  Grass- 
mann's  Analysis  will  serve  to  interest  some  to  the  extent  that 
they  may  be  led  to  investigate  his  original  works. 

The  writer  has  followed  in  tlie  main  the  notation  of  Grass- 
mann  himself,  the  principal  deviations  being  the  omission  of  the 
brackets  from  geometric  products,  writing  pq  instead  of  [pq]^ 
and  a  somewhat  different  treatment  of  the  product  p\q. 

Art.  1.     Explanations  and  Definitions. 

'The  algebra  with  which  the  student  is  already  familiar  deals 
directly  with  only  one  quaHty  of  the  various  geometric  and 
mechanical  entities,  such  as  lines,  forces,  etc.,  namely,  with 
their  magnitude.  Such  questions  as  How  much?  How  far? 
How  long?  are  answered  by  an  algebraic  operation  or  series  of 
operations.  Questions  of  direction  and  position  are  dealt  with 
indirectly  by  means  of  systems  of  coordinates  of  various  kinds. 
In  this  chapter  an  algebra  *  will  be  developed  which  deals 
directly  with  the  three  qualities  of  geometric  and  mechanical, 
quantities,  viz.,  magnitude,  position,  and  direction.  A  geomet- 
ric quantity  may  possess  one,  two,  or  all  three  of  these  prop- 
erties simultaneously;  thus  a  straight  line  of  given  length  has 
all  three,  while  a  point  has  only  one. 

The  geometric  quantities  with  which  we  are  to  be  concerned 
are  the  point,  the  straight  line,  the  plane,  the  vector,  and  the 
plane-vector. 

When  the  word  "line"  is  used  by  itself,  a  "straight  line" 
will  be  alwaj^s  intended.  A  portion  of  a  given  straight  line  of 
definite  length  will  be  called  a  "  sect  "  ;  though  when  the  length 

*  The  algebra  of  this  chapter  is  a  particular  case  of  the  very  general  and 
comprehensive  theory  developed  by  Hermann  Grassmann,  and  published  by 
him  in  1844  under  the  title  "Die  lineale  Ausdehnungslehre,  ein  neuer  Zweig 
der  Mathematik."     He  publishe  1  also  a  second  treatise  on  the  subject  in  1S62. 


EXPLANATIONS    AND    DEFINITIONS. 

of  the  sect  is  a  matter  of  indifference,  the  word  line  will  fre- 
quently be  used  instead.  Similarly,  a  definite  area  of  a  given 
plane  will  be  called  a  "  plane-sect." 

If  a  point  recede  to  infinity,  it  has  no  longer  any  significance 
as  regards  position,  but  still  indicates  a  direction,  since  all  lines 
passing  through  finite  points,  and  also  through  this  point  at 
infinity,  are  parallel.  Similarly,  a  line  wholly  at  infinity  fixes 
a  plane  direction,  that  is,  all  planes  passing  through  finite 
points,  and  also  through  this  line  at  infinity,  are  parallel.  Thus 
a  point  and  line  at  infinity  are  respectively  equivalent  to  a  line 
direction  and  a  plane  direction. 

A  quantity  possessing  magnitude  only  will  be  termed  a 
"scalar  "  quantity.  Such  are  the  ordinary  subjects  of  algebraic 
analysis,  a,  x,  sin  6,  log^,  etc.,  and  they  may  evidently  be  in- 
trinsically either  positive  or  negative. 

The  letter  T  prefixed  to  a  letter  denoting  some  geometric 
quantity  will  be  used  to  designate  its  absolute  or  numerical 
magnitude,  always  positive.  Thus,  if  Z  be  a  sect,  and  Pa  plane- 
sect,  then  TL  is  the  length  of  Z,  and  TP  is  the  area  of  P.  That 
portion  of  a  geometric  quantity  whose  magnitude  is  unity  will 
be  called  its  "unit,"  and  will  be  indicated  by  prefixing  the 
letter  U\  thus  UL  =  unit  of  Z  =  sect  one  unit  long  on  line  Z.* 
Hence  we  have  TL  .  UL  =  L. 

Art.  2.    Sum  and  Difference  of  Two  Points. 

In  geometric  addition  and  subtraction  we  shall  use  the  or- 
dinary symbols  +,  — ,  =,  but  with  modified  significance,  as  will 
appear  in  the  development  of  the  subject. 

Every  mathematical,  or  other,  theory  rests  on  certain  fun- 
damental assumptions,  the  justification  for  these  assumptions 

*  The  word  "scalar"  and  the  use  of  the  letters  T  and  U,  as  above,  were 
introduced  by  Hamilton  in  his  Quaternions,  ^stands  for  tensor,  i.e.,  stretcher, 
and  TL  is  the  factor  that  stretches  UL  into  L.  The  notation  [  L  \  for  absolute 
magnitude  is  not  used,  because  the  sign  |  has  been  appropriated  by  Grassmann 
lo  another  use. 


10  grassmann's  space  analysis. 

lying  in  the  harmony  and  reasonableness  of  the  resulting 
theory,  and  its  accordance  with  the  ascertained  facts  of  nature. 
Our  first  assumption,  then,  will  be  that  the  associative  and 
commutative  laws  hold  for  geometric  addition  and  subtrac- 
tion, that  is,  whatever  A,  B,  C  may  represent,  we  have 

A-\-B-{-C^{A-^B)-\-C=A-^{B-\-C) 

=  A-{-C+B  =  {A-\-C)  +  B. 

We  shall  also  assume  that  we  always  have  A  —  A  =  o,  and 
that  the  same  quantity  may  be  added  to  or  subtracted  from 
both  sides  of  an  equation  without  affecting  the  equality. 

Now  let/,  ,/„  be  two  points,  and  consider  the  equation 

A  +A  -  A  =  A  +  (A  -  A)  =  A-  (i> 

In  this  form  we  have  an  identity.  Write  it,  however,  in  the 
form 

A  -A  +  A  =  (A  -  A)  +  A  =  A  >  (2) 

and  it  appears  that/^  — /,  is  an  operator  that  changes/,  into 
p„hy  being  added  to  it.  Conceive  this  change  of/,  into/„  to 
take  place  along  the  straight  line  through  /,  and  />„ ;  then  the 
operation  is  that  of  moving  a  point  through  a  definite  length 
or  distance  in  a  definite  direction,  namely,  from/,  to /„.  This 
operator  has  been  called  by  Hamilton  "a  vector,"'^  that  is,  a 
carrier,  because  it  carries/,  rectilinearly  to/^.  Grassmann  gives 
to  it  the  name  Strecke,  and  some  writers  now  use  the  word 
"  stroke  "  in  the  same  sense. 

Again, /^—/j  is  the  difference  of  two  points,  and  the  only 
difference  that  can  exist  between  them  is  that  of  position,  i.e. 
a  certain  distance  in  a  certain  direction. 

Hence  we  may  regard/,  — /,  as  a  directed  length,  and  also' 
as  the  operator  which  moves  /,  over  this  length  in  this  direc- 
tion.    Writing /j  — /,  =  e,  equation  (2)  becomes 

A  +  e=A-  (3) 

*  See  the  first  of  Hamilton's    Lectures  on  Quaternions,  where  a  very  full 
discussion  of  equation  (2)  will  be  found.     Also  Grassmann  (1S62),  Art.  227. 


SUM    AND    DIFFERENCE    OF    TWO    POINTS.  11 

Thus  the  sum  of  a  point  and  a  vector  is  a  point  distant  from 
the  first  by  the  length  of  the  vector  and  in  its  direction. 

Since  /,  — /,  —  —  (/,  —/J,  it  appears  that  the  negative 
of  a  vector  is  a  vector  of  the  same  length  in  the  opposite 
direction. 

If  p^  —  px  =^0,  or  p^  =p^,p^  must  coincide  with  /,  because 
there  is  now  no  difference  between  the  two  points. 

The  question  arises  as  to  what,  if  any,  effect  the  operator 
A~/i  should  have  on  any  other  point/3,  that  is,  what  is  the 
value  of  the  expression /^ — /i+A^ 

We  will  assume  that  it  is  some  point /^,  so  that  we  have 
A— A+A=A. 
oi'  A-/.  =A-A-  (4) 

This  implies  that  the  transference  from /g  to/,  is  the  same 
in  amount  and  direction  as  that  from/, 
to  /j,  that  is,  that  /, ,  A' A' A  ^^^  ^^^^ 
four  corners  of  a  parallelogram  taken  in 
order.  Thus  equal  vectors  have  the  same  ^^^^^ 
length  and  direction,  and,  conversely, 
vectors  having  the  same  length  and  direction  are  equal. 

Note  that  parallel  vectors  of  equal  length  are  not  neces- 
sarily equal,  for  their  directions  may  be  opposite. 

Equation  (4)  may  also  be  written 

A+A=A+A.  (5) 

so  that,  whatever  meaning  may  be  assigned  to  the  sum  of  two 
points,  if  we  are  to  be  consistent  with  assumptions  already 
made,  we  must  have  the  sum  of  either  pair  of  opposite  corner- 
points  of  a  parallelogram  equal  to  the  sum  of  the  other  pair. 
The  sum  cannot  therefore  depend  on  the  actual  distances 
apart  of  the  points  forming  the  pairs,  for  the  ratio  of  these  two 
distances  may  be  made  as  large  or  as  small  as  we  please. 

If  n  be  a  scalar  quantity,  we  will  denote  that  the  operation 
e  is  to  be  performed  n  times  on  a  point  to  which  we  is  added, 
that  is,  the  point  will  be  moved  n  times  the  length  of  e;  hence 


1%  grassmann's  space  analysis. 

ne  is  a  vector  u  times  as  long  as  e,  and  having  the  same  or  the 
opposite  direction  according  to  the  sign  of  ;/. 
In  the  figure  above,  let 

A— A  =  e,,    A— A  =  e2.    A-/'i  =  e3,    A— A  =  e4- 
Then 

e>  +  e,  =A— A+A  —A  =A-A+A— A=A— A  =  ^3,  (5) 
since,  by   Eq.  4,  p,  -/,  =/^  - /^. 

Also,  e,  -  e^  ^A  -  A  =  e,.  (6) 

Hence,  if  two  vectors  are  drawn  outwards  from  a  point,  and 
the  parallelogram  of  which  these  are  two  adjacent  sides  is  com- 
pleted, then  the  two  diagonals  of  this  parallelogram  will  repre- 
sent respectively  the  sum  and  difference  of  the  two  vectors, 
the  sum  being  that  diagonal  which  passes  through  the  origin 
of  the  two  vectors,  and  the  difference  that  which  passes  through 
their  extremities.* 

Again,/,  -/,  +/3  -A  +/,  -/3  =  o  =  e,  +  e,  +  (-  ej  ; 
hence  the  sum  of  three  vectors  represented  by  the  sides  of  a 
triangle  taken  around  in  order  is  zero. 

Similarly,  if/,,  A'  •  •  -A  t>e  any  ;/  points  whatever  taken  as 
corners  of  a  closed  polygon,  we  shall  have 
(A-A)+(A-A)+(A-A)+---+(A-A-:)+(A-A)=o; 
that  is,  the  sum  of  vectors  represented  by  the  sides  taken  in 
order  about  the  polygon  is  zero.  By  "taken  in  order"  is  not 
meant  that  any  particular  order  of  the  points  must  be  obserx-ed 
in  forming  the  polygon,  which  is  evidently  unnecessary,  but 
simply  that,  when  tlie  polygon  is  formed,  the  vectors  will  be- 
the  operators  that  will  move  a  point  from  the  starting  position 
along  the  successive  sides  back  to  this  position  again,  so  that 
the  final  distance  from  the  starting-point  will  be  nothing. 

Art.  3.     Sum  of  Two  Weighted  Points. f 

Consider  the  sum  w,/',-|-;;/„/2,  in  which  m^  and  w„  are  scalars, 
that  is,  numbers,  positive  or  negative,  and  /,,  /^  are  points. 

*  Grassmann  (1S44),  §  15. 

f  Grassmann  (1844),  ^  95,  and  (1862),  Art.  227. 


SUM    OF    TWO    WEIGHTED    POINTS. 

The  scala,-s  ».  and  «.  will  be  regarded  as  values  or  weigMs 
assigned  to  the  points/,  and  A-     When  any  we.ght   s  o    un 
value  the  figure   x  will  be  omitted,  so  that  /  --  ^    ^^  ;^^^' 
called  a  unit  point.     Occasionally,  however,  a  letter  may 
used  to  denote  a  point  whose  weight  is  not  un.ty. 

To  assist  his  thinking,  the  reader  may  consider  the  weights 
initially  as  like  or  unlike  parallel  forces  acting  at  the  pomts, 

m  order  to  arrive  at  a  meaning  for  the  above  expressron 
.,e  LI  make  two  reasonable  assumptions,  wh.ch  w.ll  prove  to 
be  ct  Lrent  with  those  already  made,  viz.,  first,  that  the  sum  .s 
a  poTnt,  and  second,  that  its  weight  is  the  sum  of  the  weights 
of  the  two  given  points.     Denoting  this  sum-point  by/,  we 

*' '""  «/,/,  +  m.p,  =  ('«,  +  j".)A  ^^'^ 

Transposing,  we  have  »,(/-,  -/)  =  "'.(/  -  A).  <>" 

Both  members  of  (8)  are  vectors,  and.  being  equal,  they  must, 
by  Art.  4,  be  parallel.  This  requires  that  /  shall  be  colhnear 
wIthA  and  A.  Also,  since/.-/  and/ -A  are  v_ectors  whose 
lengths  are  respectively  the  distances  from  A  to/  and  from/ 

to  fit  follows  that  these  distances  are  in  the  rat.o  of  ,n,  to  ».. 

Hence  7  is  a  point  on  the  line  /.A  whose  distances  from/. 

and  A  -e  inversely  proportional  to  the  weights  of  these  pom  s. 

We  shall  call  7  the  mean  point  of  the  two  weighted  po.nts. 

„  „,  and  nu  are  both  positive,  (8)  shows  that /must  he     e- 

tweei,  A  andA-.  but  if  one,  say  ,«„  .s  negafve,  let  ,».—«.. 

■^'"'  «.(A-7)  =  <(A-/)'  ^9) 

and  ?  is  on  the  same  side  of  each  point,  that  is.  its  direction 
rom  each  point  is  the  same.     Also,  since  'ts  ^-stances  rom  t^ 
two  points  are  inversely  as  their  werghts,  /  must  be  nearest 
the  point  whose  weight  is  greatest. 


14  grassmann's  space  analysis. 

Case  when  w,  -j~  '"^i  =  0»  or  w,  =  —  ;;/,.* — With   this    con- 
dition  equations  (7)  and  (8)  become 

in,p,  +  inj^  =  mXp,  -  A)  =  o  .p,  ( 10) 

and  p  —p^=p—j,^_  (u) 

Thus/  is  in  the  same  direction  from  each  point,  that  is,  not 
between  them,  and  yet  is  equidistant  from  them.  This  re- 
quires either  that  the  two  points  shall  coincide,  that  is, /.^  =/i» 
which  evidently  satisfies  (10)  and  (11);  or  else,/,  and/^  being 
different  points,  that  /  shall  be  at  an  infinite  distance.  Thus 
the  sum  is  in  this  case  a  point  of  zero  w^eight  at  infinity.f 
Eq.  (10)  shows  that  a  zero  point  at  infinity  is  equivalent  to  a 
vector,  or  directed  quantity,  as  stated  in  Art.  i.  It  has  been 
shown  in  Art.  2  that  p^^p^  is  the  condition  that  /,  and  /, 
coincide;  let  us  consider  the  equalit}'  of  weighted  points  in 
general,  say  m^p^  ^=  vi„p^.  Hence,  by  (7),  there  is  found 
m^p^  —  Wj/j  =  {in^  —  m^p  =  o;  hence,  since  /cannot  be  zero, 
w,  —  ni^  =  o,  or  m,  =  in^ ;  and  therefore  w,(/,  —A)  =  O,  or, 
since  w,^o,  /,  — /„  =0,  that  is,  /,  =A-  Therefore,  if  any 
two  points  are  equal,  their  weights  must  be  the  same  and  their 
positions  identical,  that  is,  they  are  the  same  point. 

Exercise  i. — To   find    the  sum   and  difference   of   the  two 
weighted  points  3/,  and/^' 

3A+A  =  4A        3A-A  =  27, 

and  the  mean  points  are  as  shown  in 

2         _i_      _        3 ^ 

f X    "T  ;   t^i^   figure.     The  reciprocals   of   the 

-P'  ^P'    *^~'  ^'distances   of  /,  /„    and/'   from  /„, 

viz.,  i,  ^,  I",  are  in  arithmetical  progression,  hence  the  points 
form  a  harmonic  range. 

Exercise  2. — Given  a  circular  disk  with  a  circular  disk  of 

*Grassmann  (1S62),   Art.  222. 

f  Compare   the  case   of   the   resultant    of    unlike    parallel    forces    of   equal 
magnitude. 


SUM    OF    TWO    WEIGHTED    POINTS.  15 

half  its  radius  removed,  as  in  the  figure  ;  to  find  the  centroid 
of  the  remaining  portion. 

Take/,  at   center  of  large  circle, /j  at  center 
of  small  circle,  and  p^  at  the   point  of  contact ;  (       i\[   ^,    ];, 
then /g  =  !(/>,-{- /'a)-     The  areas  of  the  two  cir- 
cles are  as  i  14;  call  them  i  and  4.     Then  it  is  as 
if  there  were  a  weight  4  at  /,,  and  a  weight  —  i  at/3 ;  hence 
J  =  [4/.  -  K/,  +  A)]  -  3  =  (7/.  -  A)  -^  6. 

Prob.  I.  Show  that  /,,/,,  Jn^p^-\-  m^p^,  and  m^p^  —  7n^p^  are 
four  points  forming  a  harmonic  range. 

Prob.  2.  An  inscribed  right-angled  triangle  is  cut  from  a  circular 
disk  ;  show  that  the  centroid  of  the  remainder  of  the  disk  is  at  the 
point 

(jTT  —  2  sin  2a)  /j  —  p^  sin  2« 
3(7r  —  sin  20c)  ' 

if /i  is  the  center  of  the  circle,  p,  the  opposite  vertex  of  the  triangle, 
and  a  one  of  its  angles. 

Art.  4.    Sum  of  any  Number  of  Points. 

As  in   the  last  article  we  assume  the  sum  to  be  a  point 

whose  weight  is  equal  to  the  sum  of  the  weights  of  the  given 

points  ;  thus, 

«  « 

^mp=p27u.  (12) 

n 

Let  e  be  some  fixed  point,   and  subtract  e2m  from  both 

sides  of  (12) ;  thus  we  have 

«  « 

2jn{p  —  e)  ={p  —  e)2m,  (13) 

1  1 

an  equation  which  gives  a  simple  construction  (or  p. 

n  n 

If  2m  =  o,  then  1n^  =  —  2m,   and 

« 
2mp 

2mp  =  m,p,  +  2mp  =  m\   p,  -  -~  ],  (14) 

2m 


IG  grassmann's  space  analysis. 

so  that  the  sum  becomes  the  difference  of  two  unit  points,  or 
a  vector  whose  direction  is  parallel  to  the  line  joining  /,  with 
the  mean  of  all  the  other  points  of  the  system,  and  whose 
length  is  vi^  times  the  distance  between  these  points.  Since 
any  point  of  the  system  may  be  designated  as  /,,  it  follows 
that  the  line  joining  any  point  of  the  system   to  the  mean  of 

all  the  others  is  parallel  to  any  other  such  line.     If  '^mp  =  o, 

equation  (14)  shows  that/,  is  the  mean  of  all  the  other  points 
of  the  system,  and,  since  any  one  of  the  points  may  be 
taken  as  /,,  any  point  of  the  system  is  the  mean  of  all  the 
others. 

Let  ;/  =  3  in  (12)  and  (13);  then 

w,/,  H-  w,/,  +  m,p,  =  (;;/,  +  ;;/,  +  M,)p,  (15) 

;;/,(/.  -e)-\-  m^p-e)  +  mlp-e)={in,-^m^-^in^(p-e\     (16) 

and/  is  on  the  line  joining  the  point  w^,/,  +  w^A  ^^ith /j,  and 
therefore  inside  the  triangle  p^p^p^  if  the  ihs  are  all  positive. 
If  W3  be  negative  and  numerically  less  than  m^-\-in^,  then/ 
will  have  passed  across  the  line  /,/,  to  the  outside  of  the  tri- 
angle. If  w,  and  m^  are  negative  and  their  sum  numerically 
less  than  Wj,  then  /  will  have  passed  outside  the  triangle 
through /j,  i.e.,  it  will  have  crossed /^/j  and  /s/,.  The  point 
e  must  evidently  always  be  in  the  plane /,/,/,. 

As  a  numerical  example   let  in^  =  3,  tn^  =  4,  vi^=.  —  5,  so 
that  (16)  becomes 

p-e  =  |(/.  -e)^  2(A  -e)-  KA  -  e). 
Now,  since  e  may  be  any  point  whatever,  put   e  =  p^;    then 
'p  —  p^  =  -§.(/^  —A)  +  2(A  — A)'  ^"^  ^^1^  construction  is  shown 
in  the  figure,     p^  -  p^  =  f(/,  -A),  and  'p~p^  =  2(A  -  A)- 

As  another  example  take/  =  4/,  +  5A  ~  ^A  —  ^A,  or,  by 
(13),  making  e  =A, 

/  -A  =  4(A  -A)  +  5(A  -A)  -  2(A  -/*) 
=  A-A+A-A+7-A- 


SUM    OF    ANY    NUMBER    OF    POINTS.  17 

When  any  number  of  geometric  quantities  can  be  connected 
with  each  other  by  an  equation  of  the  form  '2nip  =  o,  in  which 
the  VIS  are  finite  and  different  from  zero,  then  they  are  said  to 
be  mutually  dependent,  that  is,  any  one  can  be  expressed  in 
terms  of  the  others.     If  no  such  relation  can  exist  between  the 


-5i' 


IV'- 


'rip, 

~Z1\      / 


quantities,  they  are  independent.     We  obtain  from  what  has 
preceded  the  following  conditions; 
That  two  points  shall  concide, 

m,p,^in^p^  =  0.  (17) 

That  three  points  shall  be  collinear, 

mj,  +  ^n^P^  +  ^>hP^  =0.  (18) 

That  four  points  shall  be  coplanar, 

;//,/>.  +  ;;/,A  +  ^^hP.  +  ^^hP.  =  O.  (19) 

It  follows  that  three  non-collinear  points  cannot  be  con- 
nected by  an  equation  like  (18)  unless  each  coefificient  is 
separately  zero.  Similarly  four  non-coplanar  points  cannot  be 
connected  by  an  equation  like  (19)  unless  each  coefificient  is 
separately  zero. 

The  significance  of  these  statements  will  be  presently  illus- 
trated. 

The  following  are  corresponding  equations  of  condition  for 
vectors : 

That  two  vectors  shall  be  parallel, 

//,€,  + ;/,e,  :=0.  (20) 


18 


GRASSMANX  S   SPACE    ANALYSIS. 


That  three  vectors  shall  be  parallel  to  one  plane, 

;/,e,  +  ;/,e,  +  ;^3e3  =  O.  (21) 

These  conditions  follow  from  the  results  of  Art.  2,  or  from 
'equations  (17)  and  (18)  by  regarding  the  e's  as  points  at  infinity. 
If  in  addition  to  (21)  we  have 

'^  +  ^h  +  ^h  =  o,  (22) 

the  extremities  of  the  three  vectors,  if  radiating  from  a  point, 
-will  be  coUinear :  for,  let  ^„  ...  ^3  be  four  points  so  taken  that 
.,?,  —  ^„  =  6, ,  ^2  ~  <^'o  =  ^2 '  ^3  ~  ^0  =  ^3 ;  then  (21)  becomes 
n,{e,  —  e^  +  nle^  —  e^  +  ^3(^3  —  e^  =  O, 

■or  by  (22)  ;//,  +  /i,e^  +  ;/3^3  =  o, 

which  by  (18)  requires  e^,  e„,  c^  to  be  collinear. 
It  may  be  shown  similarly  that 

^ne  =:  ^n  =  o  (23) 

are  the  conditions  that  four  vectors  radiating  from  a  point  shall 
have  their  extremities  coplanar. 

Exercise  3, — Given   a  triangle   e^e^e^  and   a   point  /  in    its 

plane;  pe^  cuts  e^c„  in  q^, 
pL\  cuts  e„e^,  in  q^ ,  pe„  cuts 
e,e,  in  ^,,  ^,^,  cuts  ^'/„  in /„, 
,q^q,  cuts  e^e,  in  /, ,  and  q,q^ 
cuts  e^e^  in  p„ :  to  show  that 
Pa'  Pi'  'I'lci  p„  are  collinear. 
Let/  =  ;//„  +  ;^,^,-f;/,^, ; 
jj\  then  ^,, ,  ^, ,  q^  coincide  re- 
spectively with  ;//',  -|-  n„e^ , 
n^e^-\-  n^e^,  and  n/a-\-n^€^  because  p  lies  on  the  line  joining  e^ 
with  q^ ,  etc.     Hence,  if  -r„ ,  x^,  y^ ,  j,  are  scalars, 

hence      (;r„  -y,n,)e,  -f  (x,  -j\n,)e,  -  n,(j^  -\-J\)e,  =  o. 

Now  the  ^'s  are  not  collinear,  and  yet  are  connected  by  a 


SUM    OF    ANY    NUMBER    OF    POINTS.  19 

relation  of  the  form  of  equation  (i8);  hence,  as  was  there 
shown,  each  coefficient  must  be  zero  ;  accordingly 

^0  -}\'i,  =  '^r-Io'h  =J'o  -\-J\  =  O, 

whence  we  find  x^:  x^  =  «„ :  —  «,. 

hence  (//„  —  w,)/^  =  '^o^o  ~  ^'/i »  ^"^i  similarly 

{n—n^p,  =  n,€„  —  n^t\,     {j!,  —  n^)/>^  =  n^e^  —  n^e,. 
Adding,  we  have 

('^>  -  ^i.)P.  +  kih  -  >io)P.  +  {n,  -  n)p„_  =  o; 
therefore,  by  (iS), /„ '/i  >  ^2  ^^'^  collinear. 

Exercise  4. — Let  /  =  ^Jie  -=-  2fi  be  any  point  in  the  plane- 
00 

of  the  triangle  e^e^e^ :  show  that  lines  through  the  middle 
points  of  the  sides  e^e^ ,  e^e^ ,  and  e^e^  of  the   triangle  parallel 

to  e\p,  c\p,  and  ^^p  meet  in  a  point 

P'  =  [(^^  +  ''2)^0  +  (^^2  +  ^^o)^\  +  («o  +  ^^,)^2]  -^  2:^«. 

By  the  conditions  the  vector  from  the  middle  point  of  e^e^ 
to  p'  is  a  multiple  of  the  vector  e\  — p  ;  hence 

/  -K^^  +  ^.)  =  A^o-p)     or 

P'  =  K^.  +  ^2)  +  4^0  -/)  =  i(^o  +  ^.)  +J(^2  -/), 

or,  substituting  value  of/, 

/  =  *(^,  +  ^',)  +  4^„-  -^//r-^2;/)  =  *(r„-f  ^,)+j(^,-:^;^^H-:^«).- 

hence         [(,r  —  i:)2n  -f-  7/„(j  —  ;r)]r„  +  w,(j/  —  ,t')r, 

therefore,  as  in  the  previous  exercise,  each  coefficient  must  be 
zero,  whence  ,i'  =  jf  =  -J,  and  substituting  we  find  /'  as  above^ 
It  follows  also  that  the  distances  of  p'  from  the  middle  points 
of  the  sides  are  the  halves  of  the  distances  of/  from  the  oppo^ 
site  vertices. 

2 

Prob.  3.   Show  that  e  =  ^2e  is  collinear  with  p  and  />'  of  Exer- 


20 


GRASSMANN  S    SPACE    ANALYSIS. 


cise  4.  Also  that,  by  properly  choosing/,  it  follows  that  c  is  cul- 
linear  with  the  common  point  of  the  perpendiculars  from  the  vertices 
on  the  opposite  sides,  and  the  common  point  of  the  perpendiculars 
to  the  sides  at  their  middle  points. 

Prob.  4.  Given  two  circles  and  an  ellipse,  as  in  the  figure,  with 
centers  at  <?„ ,  /j ,  and  p^.  Radii  of  circles  4  and 
I,  axes  of  ellipse  2  and  4,  small  circle  and  ellipse 
touching  large  circle  at  e^  and  <?,  respectively, 
e^e^e^  an  equihiteral  triangle:  show  that  the  cen- 
troid  of  the  remainder  of  the  large  circle,  after 
the  small  areas  are  removed,  will  be  at 

/  =  Vsli^^'o  -  A  -  2/,) =32(59^0  -  4^:  -  3^J- 

Prob.  5.   If  %  of  a  sheet  of  tin  in  the  shape 

/\  of  an  isosceles  triangle  be  folded  over  as  in 

/       \  the  figure,  show  that  its  centroid  is  given  by 

Prob.  6.   If   a   tetrahedron   e^,t\e,/^  have  a 
tetrahedron  of  ^  of  its  volume  cut  off  by  a 
plane  parallel  io  e^e/„.,  and  one  of  ■g'r  of  its 
volume  cut  off  by  a  plane  parallel   to  e/^e^ , 
show  that  the  centroid  of  the  remaining  solid  is  at 

/=  880(227^0  +  175^3  +  239(^1  +  O  )• 


\ 


Art.  5.     Reference  Systems. 


Let  p  be  any  unit  point,  e,,  e,,  e^  three  fixed  unit  points, 
and  w,  x,y  scalars  ;  then,  writing 

p  =  tve,-\-xe,-^ye.,,  '  (24) 

we  must  have  also,  because/  is  a  unit  point, 

w  ^  X  ^  y  =  I,  (25) 

and  /  is  the  mean  of  the  weighted  points  we^,  xe,,ye,.  The 
point/  may  occupy  any  position  whatever  in  the  plane  e^i\e^; 
for  it  is  on  the  line  joining  zvr,  +  xe^  with  e„,  and  by  varying 

y  and  w -\- x,  —  remaining  constant,  /  may  be  moved  along 
X  ^ 


REFERENCE    SVSTEM3.  21 

this  line  from  —  oo  to  -|-  co  ;  while  by  varying  the  ratio  —  the 

X 

point  we^-\- xe^  may  be  moved  from  —  co  to  -f-  "^  along  e^e^, 
and  thus  the  first  line  will  be  rotated  through  i8o  degrees,  and 
p  may  thus  be  given  any  position  whatever  in  the  plane. 

A  system  of  unit  points  to  which  the  positions  of  other 
points  may  be  referred  is  called  a  reference  system,  and  the 
triangle  e^e^e^  is  a  reference  triangle.  For  reasons  that  will  ap- 
pear  later,  the  double  area  of  this  triangle  will  be  taken  as  the 
unit  of  measurement  of  area  for  a  point  system  in  two-dimen- 
sional space. 

Similarly,  in  solid  space,  taking  a  fourth  point  e^,  we  write 

p  =  zve,  -f-  xt\  +  jr,  +  ^^3,  (26) 

which  implies  also       iv  ^  x  -\-  y  -\-  z  ^  i  ;  (27) 

and  p  may  be  shown  as  above  to  be  capable  of  occupying  any 
position  whatever  in  space  by  properly  assigning  the  values  of 
zv,  X,  y,  z\  so  that  ^„,  .  .  .  e^  form  a  reference  system  for  points 
in  three-dimensional  space.  The  tetrahedron  e^e^e^t\  is  called 
the  reference  tetrahedron,  and  six  times  its  volume  will  be 
taken  as  the  unit  of  volume  for  a  point  system  in  three-dimen- 
sional space. 

Eliminating  zu  between  (24)  and  (25),  we  have 

/  =  ^'0  +  '<^,  -  ^'„)  +  jC^,  -  e^,  (28) 

from  which  it  may  also  be  easily  seen  that/  may  be  any  point 
in  the  plane  e^e^e^.  Writing/  —  e^  =  p,  e^-  e,  =  e,,  c^_  —  e,-  e„ 
(28)  becomes  p  =  xe^  +  ye, ,  (29) 

and  e,,  e,  form  a  plane  reference  system  for  vectors. 
Similarly,  from  (26)  and  (27)  we  find 

pz=  A-e. +/e, +  ^€3,  (30) 

and  e,,  e^,  e,  are  a  reference  system  for  vectors  in  solid  space, 
any  vector  whatever  being  expressible  in  terms  of  these 
three. 

If,  in  equations  (29)  and  (30),    the  reference  vectors  are  of 


22  grassmann's  space  analysis. 

unit  length  and  mutually  perpendicular,  we  have  unit,  normal 
reference  systems,  and  in  this  case  i. ,  i^,  i^  will  generally  be  used 
instead  of  e,,  e^,  e^. 

Exercise  5. — To  change  from  one  reference  system  to  an- 
other, say  from  e^,  e^,  e^  to  ^;',  r/,  e^. 

The  new  reference  points  must  be  connected  with  the  old 
ones  by  equations  such  as 

c-,  =  '^^o'  +  ^^//  +  '^/Z- 
Then  any  point/  =  x^c^  -\-  x^e^  -f-  x^e^  will  be  expressed  in 
terms  of  the  new  reference  points  by  substituting  the  values  of 
t\,  etc.,  as  given.  If  e^,  ^/,  e^  are  given  in  terms  of  the  old 
points,  ^0,  e^,  c  may  be  found  by  elimination.  Thus,  if  r„'  =  ^/.f, 
e'  =  2!/ie,  e^  =  ^ne,  we  have  at  once 


h 

/, 

K 

e: 

/. 

h 

m. 

;//, 

m. 

^0  = 

e: 

w/j 

in^ 

^',> 

'^ 

n. 

e: 

'>i. 

n. 

with  similar  values  for  e^  and  e^. 

As  a  numerical  example  let  the  new  reference  triangle  be 
formed  by  joining  the  middle  points  of  the  sides  of  the  old  one. 
Then  e^  —  \{c,  -f  rj,  e^  =  i(^„  +  ^„),  ^/  =  ^{e,  +  e^) ;  whence 
^0  =  -  ^0'  +  e!  +  e^,  e,  =  e^  -  e/  +  e,',  e,  =  e,'  +  ^/  —  ^./. 
Thus  p  =  .r„r„  +  ,r,r,  +  ,r/., 

=  (-  '^'o  +  '^'i  +  '^'.)^o'  +  U'o  -  'I',  +  -l-',)^/  +  (-S^o  +^^  —  ^.)^/- 

Exercise  6. — Three  points  being  given  in  terms  of  the  refer, 
ence  points  e^,  i\,  r„,  find  the  condition  that  must  hold  between 
their  weights  when  they  are  collinear. 

2  2  2 

Let  />„  =  2/c,  /i  =  ^;ne,  p^  =  '2ne\  then,  k^,  ^,,  k^   being 
000 

scalars,  we  must  have  for  collinearity.  by  (18), 


(3i> 


REFERENCE    SYSTEMS.  33. 

that  is,  k^'2le -\-  k'E^^ne  -\-  k^ne  =  o, 

whence         {kj^  +  k^7;i^  +  /^2«o)^o  +  (^'oA  +  ^'i^«,  +  ^\«i)^i 

and,  as  e^,  e^,  l\  are  not  colhnear,  the  coefficients  must  be  zero, 
by  Art.  4 ;  hence 

kj,  +  k^m^  +  k^n^=  kj^  -f  /i-,///,  +  k^n^  =  kj^  +  /',;;/,  +  k.^n^  -  o, 
and,  by  ehmination  of  the  k's, 

/„     m,     11, 

which  is  the  required  condition  of  colhnearity. 

Prob.  7.  If  />  =  3^0  -  ^,  —  ^2 .  4^«'  =  Zt\  +  ^2 '  4^:'  =  3^2  +  ^0 , 
4''/  =  y,  +  ^1 ,  show  that  7/  =  —  19^'/  —  y\'  +  29^/. 

333  3 

Prob.  8.   Find  the  condition  that  four  points  ^ke,  "Sle,  2i}ie,  2ne- 

0000 

shah  be  cophxnar.     Ans.   [/('„ ,  /j ,  ///, ,  n^]  =  o. 

Prob.  9.  li p  ^  7ue^  -\- xe^ -{- ye„,  and  there  exist  between  the 
scalars  70,  x,y  a  hnear  relation  such  as  Atv  +  Bx  -\-  Cy  =  o,  A,  B, 
C  being  scalar  constants,  show  that/ will  always  lie  on  a  straight 
line  which  cuts  the  reference  lines  in  Ae^  —  Be^ ,  Ae^  —  Ce^ ,  and 
Cfj  —  Be^.  Consider  the  special  cases  when  A  =:  B,  B  ^=  C,  C=  A, 
A  =  B  =  C,  A  =  o,  B  =  o,  and  C  =  o. 

Prob.  10.   If/  =  7i'e„  +  Xi\  -\- ye„_  +  se^,  and  there  exist  also  an 

equation  A7o  -\-  Bx  -{- Cy  -\-  Dz  =  o,  show  that/  will  lie  on  a  plane 

c  c 

which    cuts    the    edges    of    the    reference   tetrahedron    \\\   -^ ,", 

°  B        A' 

e         e 

—; -J-,  etc.      Also,   if    a  second   relation   between   the    variables, 

such    as   A'tv  +  B' x  +  Cy  -\-  D'z  =  o,  be    given,  then  /  lies  on  a 
line  which  pierces  the  faces  of  the  reference  tetrahedron  in 


^0  ^,  ^. 
ABC 
A'    B'    a 


e        c        e 
DAB 
D'     A'     B' 


etc. 


'24  grassmann's  space  analysis. 

Art.  6.     Nature  of  Geometric  Multiplication* 

The  fundamental  idea  of  geometric  multiplication  is,  that  a 
product  of  two  or  more  factors  is  that  which  is  determined  by 
those  factors. 

Thus,  two  points  determine  a  line  passing  through  them, 
and  also  a  length,  viz.,  the  shortest  distance  between  them  ; 
hence /j/',  —  L  \s  the  sectf  drawn  from  p^  to/^i  or  generated 
by  a  point  moving  rectilinearly  from /^  to/^. 

The  student  should  note  carefully  the  difference  between 
p^p^  and/^  — /"i ;  they  have  the  same  length  and  direction,  but 
the  sect  p^p^  is  confined  to  the  line  through  these  two  points, 
while  the  vector/,  —  />,  is  not.  The  sect  has  position  in  addi- 
tion to  the  direction  and  length  possessed  by  the  vector. 

Again,  in  plane  space,  two  sects  determine  a  point,  the 
intersection  of  the  lines  in  which  they  lie,  and  also  an  area,  as 
will  appear  later,  so  that  L^L^  —  Pi  i'l  which  p  is  not  in  general 
a  unit  point.  In  solid  space,  however,  two  lines  do  not,  in 
general,  meet,  and  hence  cannot  fix  a  point ;  but  two  sects,  in 
this  case,  determine  a  tetrahedron  of  which  they  are  opposite 
edges. 

It  appears,  therefore,  that  a  product  may  have  different 
interpretations  in  spaces  of  different  dimensions.  Hence  we 
will  consider  separately  products  in  plane  space,  or  planimetric 
products,  and  those  in  solid  space,  or  stereometric  products. 

Products  of  the  kind  here  considered  are  termed  "  com- 
binatory," because  two  or  more  factors  combine  to  form  a 
new  quantity  different  from  any  one  of  them.  This  is  the 
fundamental  difference  between  this  algebra  and  the  linear 
associative  algebras  of  Peirce,  of  which  quaternions  are  a 
special  case. 

Before  discussing  in  detail  the  various  products  that  may 
arise,  we  will  give  a  table  which  will  serve  as  a  sort  of  bird's-eye 
view  of  the  subject. 

*  Grassmann  (1S44),  Chap.  2  ;  (1S62),  Chap.  2. 
f  See  Art.  i. 


NATURE    OF    GEOMETRIC    MULTIPLICATION. 


25 


In  this  table  and  generally  throughout  the  chapter  we  shall 
wsQ  p,  p^,  p^,  etc.,  for  points;  e,  e^,  e^,  etc.,  for  vectors  ;  L,  L^, 
etc.,  for  sects,  or  lines;  ?/,  z/^,  etc,,  for  plane-vectors  ;  and  P,  P^, 
etc.,  for  plane-sects,  or  planes.  Also p, p^,  etc.,  as  used  in  this 
table  will  not  generally  be  unit  points. 

The  products  are  arranged  in  two  columns,  so  as  to  bring 
out  the  geometric  principle  of  duality. 

Planimetric  Products. 


/lAA  =  'ire'^  (scalar). 

pL  =  area  (scalar). 

/,  •  ^,4  =  L. 

P.P.- P.P.  ^P- 

P.P^  -P.P.  •/'r,/'o=^  (area)Xscalar). 


L,L,=p. 

L^L^L^  =  (area)''(scalar). 

Lp  =  area  (scalar). 

A-/,A  =  A 
L^L^.L^L,.L,L,=  (area)'(scalar) 


e^e^  =  area  (scalar). 


Stereometric  Producis. 


P.P.  =  J- 

P,P,  ^-  L. 

P.P.P.  =  P- 

P.P.P.^P- 

PiP.P^P^  =  volume  (scalar). 

P,P„P,P,  =  (volume)'  (scalar). 

pP  =  volume  (scalar). 

Pp  —  volume  (scalar). 

L^Li  =  volume  (scalar). 

L^L^  =  volume  (scalar). 

pL  =  Lp  =  P. 

PL  =  LP  =  p. 

p.PA  =  P. 

P'P.P.=P- 

p.PAP,=L. 

P.p,p„j,  =  L. 

L-p.p,P,=p. 

L.P,P,P,  =  P. 

€.e,  =  ri. 

V.V,  =  e. 

€,€^63  =  volume  (scalar). 

7j ^11^11^  =  (volume)'  (scalar). 

^,e, .  636,  =  e. 

VJh-V,V.  =  V- 

26  grassmann's  space  analysis. 

Laws  of  Combinatory  Multiplication.  —  All  combinatory 
products  are  assumed  to  be  subject  to  the  distributive  law  ex- 
pressed by  the  equation 

A{B  -{-  C)  =  AB  ^  AC. 

The  planimetric  product  of  three  pomts  or  of  three  lines, 
and  the  stereometric  product  of  three  points  or  planes,  or  of 
four  points  or  planes,  are  subject  to  the  associative  law.  That  is, 

In  Plane  Space  : 
AAA  =AA-A=A-AA;    AA4  =  L^L^.L^  =  L^  •  L^L^. 

In  Solid  Space  : 
P. P.P.  =P.-P.P.=  P.P.  A ;    P'.P.P.  =  P. ■  P.P.  =  P.P. •  ^3. 

P.P.P.P^=P.-P.P.P^  —P.P.-PzP.'^ 

PPPP=P.PPP=PP    PP, 

The  commutative  law  of  scalar  algebra  does  not,  in  general, 
hold.  Instead  of  this,  in  the  products  just  given  as  being  asso- 
ciative, a  law  prevails  which  may  be  expressed  by  the  equation 

AB=  -BA, 

from  which  it  follows   that  the  interchange  of  any  two  single 
factors  of  those  products  changes  the  sign  of  the  product.* 

Since  vectors  are  equivalent  to  points  at  co ,  the  associative 
law  holds  for  e^e^e^  and  rjji^i]^- 

Art.  7.    Planimetric  Products. 

Product  of  Two  Points.f— This  has  been  fully  defined  in' 
Art.  6,  and  it  is  evident  from  its  nature  as  there  given  that 

AA=-AA-  (32) 

If  p^  z=/>„  this  becomes  pj^  =  o,  which  must  evidently  be 
true,  since  the  sect  is  now  of  no  length. 

Also,  A(A  -A)  =  AA  - AA  =  AA-  (33> 

*  Grassmann  (1862),  Chap.  3.         f  Grassmann  (1S62).  Arts.  245,  246,  247. 


PLANIMETKIC    PRODUCTS. 


27 


But/j  — Px  is  a  vector,  say,  e  ;  hence 

P,e=PxP-,'^  (34) 

■lor  the  product  of  a  point  and  a  vector  is  a  sect  having  the  di- 
rection and  magnitude  of  the  vector  ;  or,  again,  multiplying  a 
vector  by  a  point  fixes  its  position  by  making  it  pass  through 
the  point. 

To  find  under  what  conditions//''  will  be  equal  to  p^p^. 
Take  any  other  point /j  in  the  plane  space  under  consideration, 
and  write  p  =  x,p,  +  x^^-^x,p,,  p'  =y,p,  +y,p..+y,p,,  with 
the  conditions  for  unit  points  2x  =  2y  =  o. 


Then      pp'  = 


,r,  X, 

J,   J'. 


AA  + 


x^  x^ 

AA  + 

x^  x^ 

J,  J'3 

Jz    J> 

PzPv 


If  this  is  to  reduce  to p^p„,  we  must  have  the  third  condition 
-^2^3  ~  '^'3^2  =  '^'3^1  ~  '^'1 J3  =  o>  which  requires  that  x\  =  j^  =  o, 
unless  the  coefificient  of  p^p^  is  to  vanish  also.  Thus  pp'  must 
be  in  the  same  straight  line  with/,/^-  If,  moreover,  in  addition 
-^J'2  —  x^y^  =  I'  we  shall  have//'  =  p,p^.  Hence  //'  is  equal 
to/,/^  when,  and  only  when,  the  four  points  are  collinear,  and 
/'  is  distant  from/  by  the  same  amount  and  in  the  same  direc- 
tion that/2  is  from/,. 

Product  of  Three  Points. — By  Art.  6  the  product  is  what 
is  determined  by  the  three  points.  In  solid  space  they  would 
fix  a  plane,  but,  as  we  are  now  confined  to  plane  space,  this  is 
not  the  case.  The  points  evidently  fix  either  a  triangle  or  a 
parallelogram  of  twice  its  area,  and  the  product  p,p„p^  will  be 
taken  as  the  area  of  this,  or  an  equivalent,  parallelogram. 

This  area  is  taken  rather  than  that  of  the  triangle,  because 
it  is  what  is  generated  by/,/,  as  it  is  moved  parallel  to  its 
initial  position  till  it  passes  through /j. 

We  have  p^P4>,  =/,  .p,p,  =  -/,  .p,p,  =  -  pj,p„  so  that 
if  we  go  around  the  triangle  in  the  opposite  sense  the  sign  is 
changed.  As  this  product  possesses  only  the  properties  of  mag- 
nitude and  sisfii  it  is  scalar. 


Write  /  =  2xp,  /'  =  ^>/,  /"  =:  :Ssp  ;  then 


28  grassmanin's  space  analysis. 


///'  = 


1  2  3 

J,  j»  yz\p.P.P*'^  (35> 


that  is,  any  triple  point  product  in  plane  space  differs  from  any 
other  only  by  a  scalar  factor.* 

Finally,    AAA  = //A  -  A)(A  -  A)  =  A^e',  (S^) 

if  e  =  A  — /.  and  e'  ^^^  — /,. 

Product  of  Two  Vectors. — Using  the  values  of  e  and  e' 
just  given,  we  see  that  e  and  e'  determine  the  same  paral- 
lelogram that  /,,  A,  and  p^  do;  hence  the  meaning  of 
the  product  is  the  same  in  all  respects  in  two-dimensional 
space. 

We  shall  have  ee'  =  —  e'e,  for 

ee'  =  (A-A)(A-A)=  -(A-A)(A-A)=  -e'e; 

since  we  have  shown  that  inverting  the  order  changes  the  sign 
in  a  product  of  points.  The  result  may  be  obtained  also  by 
regarding  e  and  e'  as  points  at  infinity,  or  by  consideration  of 
a  figure. 

As  we  have  seen  that  ee'  has,  in  plane  space,  precisely  the 
same  meaning  3.s p^p^p^  we  may  write 

P.P.P.^P.^^'  ^  ee' 

=  (A  -/.)(A  -A)  =P.P.+P^P^+P.P.'  {Z7) 

Thus  the  sum  of  three  sects  which  form  the  sides  of  a  triangle, 
all  taken  in  the  same  sense  as  looked  at  from  outside  the 
triangle,  is  equal  to  the  area  of  the  triangle. 

Product    of    Two    Sects. — Any  two    sects    in  plane  space,. 
Z,,  Z,,  determine  a  point,  the  intersec- 
/    tion  of  the  lines  in  which  they  lie,  and 

an    area,  that  of  a  parallelogram  as  in 

T '"P\ 

'  the  figure.     Let  /„  be  the  intersection, 

and  take  /,  and  ^  so  that  Z,  =/„/■  ^""^  A  =  AA-     The  area 
*  Grassmann  (1S62),  Art.  255. 


PLANIMETRIC    PRODUCTS.  29- 

determined   by    L^  and  L^    is    then    the    same    that    we    have 
given  as  the  value  oi p^p^p^.     We  write  therefore 

^:A   =  A/:  'P.P.  =  A/.  A  •  A-  (38) 

The  third  member  of  (38)  is  not  to  be  regarded  as  derived 
from  the  second  by  ordinary  transposition  and  reassociation  of 
the  points,  for  the  associative  law  does  not  hold  for  the  four 
points  taken  together,  since  A/i A  -A  =  O-  ^^^^  third  member 
simply  results  from  the  definition  of  Z,Zj.*  It  may  be  taken 
as  a  model  form  which  will  be  found  to  apply  to  several  other 
cases,  for  instance  to  (38)  when  points  and  lines  are  inter- 
changed throughout.     Thus,  if/j  =  Z„/.,  and/,  =  L^L^we  have 

/:  A  =  L,L^  •  L,L„_  =  L^L^L^ .  Z„.  (39) 

For  take//  and //so  that/,//  =  Z,  and/^//=  L,]p^p^  is 
evidently  some  multiple  of  L^  ,  say  nL^ ;  hence 

AA  =  «^o  =  ;^( AA  •  AA')  •  (AA  •  AAO 

=  ^.(AA/,'-/,)  •  (/.AA'  -A),  by  (38), 

=  —2  •  /.A/i'-AAA'  •  /.A.    because  /,/,//  and 

PiPiPi  ^^'^  scalar, 
=   '-■  (P.P.  •  /■//  •  AA')  •  A-  by  (38), 
=  L^L^L„ .  Z„ ,  which  was  to  be  proved. 

Product  of  Three  Sects. — The  method  has  just  been  indi- 
cated, but  we  may  also  proceed  thus:  Let  the  lines  be 
Z„,  Z,,  Zj,  and  let/^,/,,/„  be  their  common  points.  Take 
scalars  «„,  ;;,    ?i^  so  that  Z„  =  n^p^p^,  etc.,  then 

L,L,L^  =  nji^n, .  p,p,- p.p^.  p„p,  =  -  n,n,n^ .  p^p^  .p^p^  .p^p^ 

=  -  n,n,n^  •  A/,A  •  A/o/>  =  '^o'^>«.(/',A,A)'-  (40) 

*  Grassmann  applies  the  terms  "eingewan  it  "  and  "  regressiv  "  to  a  prod- 
uct of  this  kind,  the  first  term  being  used  in  the  Ausdehnungslehre  of  1844, 
and  the  second  in  that  of  1S62.  See  Chapter  3  of  the  first,  and  Chapter  3„ 
Art.  94,  of  the  second. 


30  grassmann's  space  analysis. 

Product  of  a  Point  and  Two  Sects, — Let/  be  any  point  and 
let  Z,  and  Z,  be  as  in  (38) ;  then 

pL,L^  =p.p,p,  .pj,  =P  -PoPJ^  -A  =PoP,P^-PP,'      (41) 

It  has  been  here  assumed  that  pL^L^  =/  .  LJ^„.  The  prod- 
uct is  not  associative,  for  pL^  .  L^  is  the  line  Z„  times  the 
scalar /Zj,  a  different  meaning  from  that  assigned  in  (41).  As 
a  rule,  to  avoid  ambiguity,  the  grouping  of  such  products  will 
be  indicated  by  dots. 

Product  of  Two  Parallel  Sects, — Let  them  be/je  and  np^e\ 
then,  as  in  (38), 

/,e .  np^e  =  n  .p^e  .p„e  =  n  .  ep, .  e/,  =  n  .  ep,p^  .  e,       (42) 
that  is,  a  scalar  times  the  common  point  at  00 . 

Addition  and  Subtraction  of  Sects. — Let  Z,  and  Zj  be  two 
sects,  pg  their  common  point,  and  p^  and  p^  so  taken  that 
A  =PoP^,  A=AA;  then 

Z,  +  Z,  =/„/>,  +AA  =  A(/:  +A)  =  2AA  (43) 

/  being  the  mean  of/,  and/„;  hence  the  sum  is  that  diagonal 
of  the  parallelogram  which  passes  through /„.     Also 

A- A=A(A-A)>  (44) 

so  that  the  difference  of  the  two  passes  also  through /„  and  is 
parallel  to  the  other  diagonal  of  the  parallelogram  determined 
by  Zj  and  Z„. 

If  the  two  sects  are  parallel  let  them  be  w,/,e  and  n„p„e\ 
then 

«,/,€  +  n^p^e  =  (;/,A  +  '^.A)e  =  {».  +  ».)pe,,         (45) 

so  that  the  sum  is  a  sect  parallel  to  each  of  them,  having  a 
length  equal  to  the  sum  of  their  lengths,  and  at  distances  from 
them  inversely  proportional  to  their  lengths. 

If  «,  =  —  «,  the  two  sects  are  oppositely  directed  and  of 
equal  length,  and  the  sum  is 

nXp.e  -p^e)  =  n^p,  - P,)e,  (46) 

which,  being  the  product  of  two  vectors,  is  a  scalar  area. 


PLANIMETRIC    PRODUCTS.  31 

Consider  next /^  sects /,e, , /^e^ ,  .  .  .  pne,,,  and  let  c^  be  some 
arbitrarily  cliosen  point;  tlien 

n  n  11  n  n  n 

^p€  E  e,^e  -  e^'Se  +  ^pe  =  e^:^^e  -f-  ^{p  -  ^Je.  (47) 

Tlie  second  term  of  the  tliird  member  of  tliis  equation,  being  a 
sum  of  double  vector  products,  that  is,  a  sum  of  areas,  is  itself 
an  area,  and  is  equal  to  the  product  of  any  two  non-parallel  vec- 
tors of  suitable  lengths.  Therefore,  a  and  (3  being  such  vec- 
tors, write  ^'e  =  a  and  2{p  —  i\)e  =  aft.  Hence  (47)  become 
^pe  =  cy,  +  a'/J  =  {e,  -  fS)a.  (48) 

Let  q  be  some  point  on  the  line  ^pe\  then 
g^pe  =  O  =  qt\a  +  ga(3  ^  qt\a  +  Lxf3, 
by  (37),  hence     qi\cx  =  —  a(3  =  f3a. 

The  figure  presents  the  geometrical  mean- 
ing of  the  equation,  and  hence  it  appears  that 
qa{=^  ^pe)  is  at  a  perpendicular  distance  from 
e,  of 

cy£__^p-  Oe 

Ta  ~         T^e       ' 


(49) 


It  is  easily  seen  that  a  sect  possesses  the  exact  geometrical 
properties  of  a  force,  namely,  magnitude,  direction,  and  position, 
and  the  discussion  of  the  summation  of  sects  which  has  just 
been  given  corresponds  completely  to  the  discussion  of  the  re- 
sultant of  a  system  of  forces  in  a  plane.  In  this  algebra,  then, 
the  resultant  of  any  system  of  forces  is  simply  their  sum,  and 
this  will  be  found  hereafter  to  be  equally  true  in  three-dimen- 
sional space.  The  expression  in  (46)  corresponds  to  a  couple, 
as  does  also  the  ^X/  —  ^'u)^  of  (47);  '^"^  t^'i's  equation  proves 
the  proposition  that  any  system  of  forces  in  a  plane  is  equiva- 
lent to  a  single  force  acting  at  an  arbitrary  point,  ^,,,  and  a 
couple.  Equation  (49)  gives  the  distance  of  the  resultant  from 
this  arbitrary  point. 

Exercise  7. — To  find  x,y,  .0  from  the  scalar  equations 


32 


GRASSMANN  S    SPACE    ANALYSIS. 


Multiply  the  equations  by  p^,  p^,  and  p^  respectively,  and 
add  ;  hence 

3  3  3  3 

x:^ap  +  }>:^hp  -\-  z:^cp  —  :^dp. 
1  111 

Now  '^ap,  ^bp,  etc.,  are  points  :  multiply  the  equation  just 
written  by  '^ap.'^bp;  thus 

z'^ap  .  ^bp:^cp  =  '2ap  .  :^bp .  '2dp, 
because  "^ap  .  2ap  =  o,  etc.;  therefore 
z  =  ^ap .  2bp .  :^dp  -^  :^ap .  ^bp^cp  =  [a, ,  b„^ ,  d,]  H-  [a, ,  b„_ ,  c,] , 

a  very  simple  proof  of  the  determinant  solution.  Of  course 
X  and  }'  will  be  found  by  multiplying  by  the  other  pairs  of 
points. 

Exercise  8. — Forces  are  represented 
by  given  multiples  of  the  sides  of  a  par- 
allelogram ;  determine  their  resultant. 

Let  the  parallelogram  be  double  the 
triangle  <?//„,  and  the  forces 

k,e,e,  +  k,e,(/,  -  ^J  +  k„/„_{c,  -  ^.)  +  k,c,e,  =  2pe 

=  {/c;  +  Z^).'/.  +  (/'.  +  k,y,e,  +  {k,  +  /c,)e,e. 

Multiply  by  e^e^  to  find  where  the  resultant  cuts  tiiis  line  , 
then 

or  r/,  cuts  the  resultant  at  the  point 

[(^',  +  ^%k  -  (^'.  + /^sVo]  -  (^^  -  ^%)- 

Similarly  the  resultant  cuts  the  other  sides  of  the  reference 
triangle  at  [(k^  +  k,y^  —  (/■„  +  ^,)^J  h-  (4  +  ^^  —  ^'o  —  ^\)  and 
at  [(X'„  +  X'.).„  -  {k,  +  k,\-]  -  {k^  -  K). 

Suppose  X%  =/(',:=/'„  = /'j ;  then  each  of  the  three  points 
just  found  recedes  to  infinity;  but  in  this  case  ^pe  reduces  to 
2kle,e^  +  ^/,+  ^,0  =  2^„(^,  —  eX^^  —  ^.),  and  the  system  is 
equivalent  to  a  couple. 

Prob.  II.  Construct  the  resultant  of  Exercise  8  when  /'„  ==  i, 
/^,=  2,  k.^r=  3,  k^=  4;  when  /.\=  i,  l\—  —  2,  k^=  3,  /&,=  —  4;  when 
-^0  —  3>  ^1  —  -^3  —  2,  /^2  =  I ;  'incl  wlien  k^  =  >^^  =  i,  ^'j,  =  /^^  =  —  2. 


THE    COMPLEMENT.  33 

Prob.  12.  There  are  given  ;/  points  /,  .  .  ./„;  to  find  a  point  e 
such  that  forces  represented  by  the  sects  <f/j ,  cp^,  etc.,  shall  be  in 

equilibrium.      (The  equation  of  equilibrium  is  ^ep^e^p^—ep  =  o. 

Hence  e  coincides  with  the  mean    point  of  the/'s.) 

Prob.  13.  If  a  harmonic  range  e^,  p,  e.^,  p'  be  given,  together  with 
some  point  e^  not  coUinear  with  these  points,  show  that 

^0'',  /  •  e,e,j'  =  —  e^  pe^  .  e.p'e^. 
(Let  p  =  ;;;/',  +  »i^c^   and  p'  =  w/,  —  w^'^  >    ^^    ^"    Exercise    2  of 
Art.  3.) 

Prob.  14.  Show  that  the  relation  of  Prob.  13  holds  for  any  four 
points  whatever  taken  respectively  on  the  four  lines  e/^,  e^p,  e/.^, 
e^p'.  If  the  four  points  are  all  at  the  same  distance  from  e^,  show 
that  the  areas  e^e^p,  etc.,  become  proportional  to  the  sines  of  the 
angles  between  e/^  and  i\p,  etc. 


Art.  8.    The  Complement.* 

Taking  point  reference  systems,  or  unit  normal  vector  ref- 
erence systems,  as  in  Art.  5,  the  product  of  the  reference  units 
taken  in  order  being  in  any  case  unity,  the  complement  of  any 
reference  unit  is  the  product  of  all  the  others  so  taken  that 
the  unit  times  its  complement  is  unity. 

To  find  the  complements  of  quantities  other  than  reference 
units  the  following  properties  are  assumed  : 

{a)  The  complement  of  a  product  is  equal  to  tlie  product 
of  the  complements  of  its  factors. 

{b)  The  complement  of  a  sum  is  equal  to  the  sum  of  the 
complements  of  the  terms  added  together. 

{c)  The  complement  of  a  scalar  quantity  is  the  scalar  itself. 

Considering  now  the  point  system  in  plane  space  f„,  r,,  e^ 
with  the  constant  condition  ^//^  =  i,  the  sides  of  the  refer- 
ence triangle  taken  in  order  are  the  complements  of  the  oppo- 
site vertices,  and  vice  versa. 

The  complement  of  a  quantity  is  indicated  by  a  vertical 
line,  as  \p,  read,  complement  of/. 

*  See  Ausdehnungslehre  of  1862,  Art.  8g. 


34  grassmann's  space  analysis. 

Thus  k„  =  ^A,       k/,  =  l(ko)  =  ^o» 

k2  =  ^«^,,       k„^.  =l(i^O  =  ^,- 

For  rj^o  =  ^//,  =■  I,  which  agrees  with  the  definition  ; 

V,c^  =  V,-V^  =  i\i\  .  e,t\  =  —  e^c^ .  e,e^  =  -  ^//.  .e^  =  e„  by  {a) 

and  (38) ; 

ka^'/.=  ko  •\e,-\e^  =  ^/.-  ^.^0  •  ^0^1  =  (^o^/J'  =  I  =^0^/.,  which 
agrees  with  (^)  ;  e^[c^  =  ^//„  =  o  =  rj^,  =  e^\e^. 

Next  take  any  point /j  =  ^/r,  and  we  have,  by  {b), 

0 

lA=^v|.=/„./,+/..,r„+4.„.,=/y.4^^|  - ^"jg - ^»j  =  L,  (50) 

Thus  the  complement  of  a  point  is  a  hne,'^  which  may  be 
easily  constructed  by  the  fourth  member  of  (50),  winch  ex- 
presses this  line  as  the  product  of  the  points  in  which  it  cuts 
the  sides  e/^  and  ^/,  of  the  reference  triangle.  Comparing 
this  equation  with  Ex.  3  in  Art.  4,  it  appears  that  ]/,  above  is 

related  to  the  point  ^  %  as  the  line/^/j  of  Ex.  3  is  to  the  point 

'2)ic.      Hence  \p^  may  be  found  by  constructing  this  line  cor- 

1  g 
responding  to  ^ -,  as  shown  in  the  figure  of  Ex.  3,  Art.  4. 

0  / 

Again,  the  line  |/,  may  be  shown  to  be  the  anti-polar  of  p 
with  respect  to  an  ellipse  of  such  dimensions,  and  so  placed 
upon  ^//„  that,  with  reference  to  it,  each  side  of  the  reference 
triangle  is  the  anti-polar  of  the  opposite  vertex.*  F'rom  this 
it  appears   that  complementary  relations  are  polar   reciprocal 

relations.     Take  any  point/".,  =  '^me,  and  we  have 

"0 

=  .iVw  =  "^inc .  ^'/\  £  =  /.,  I/,,  (51) 

0 

*See  Hyde's  Directional  Calculus,  Arts   41-43  and  121-123. 


THE    COMPLKMENT. 


35 


so  tluit  tliis  product  is  commutative  about  the  complement 
sign,  and  scalar.  This  is  true  of  all  such  products  when  the 
quantities  on  each  side  of  the  complement  sign  are  of  the  same 
order  in  the  reference  units.  Take  for  instance  the  product 
/lAlAA-  This  is  scalai  because  \p^p^  is  a  point,  so  that  the 
whole  quantity  is  equivalent  to  a  triple-point  product ;  and  we 

have/,/,  I/3A  =  lAA  -/.A  =  I  KP^P^  l/iA)  =  A/4  i/iA.  by  {a)  and 
(ct).  If,  however,  such  a  quantity  be  taken  2iSp^p^ .  \p^  it  is  neither 
scalar  nor  commutative  about  the  sign  |  ;  for,  \p^  being  a  line, 
the  product  is  that  of  two  lines,  that  is,  a  point,  and 

/.A  •  I A  =  -  I A  -/.A  =  -  I  (A  •  l/,A)-  (52) 

Such  products  as  we  have  just  been  considering  are  called 
by  Grassmann  "inner  products,"*  and  he  regards  the  sign  | 
as  a  multiplication  sign  for  this  sort  of  product.  Inasmuch, 
however,  as  these  products  do  not  differ  in  nature  from  those 
heretofore  considered,  it  appears  to  the  author  to  conduce  to 
simplicity  not  to  introduce  a  nomenclature  which  implies  a  new 
species  of  multiplication.  For  instance,/]^  will  be  treated  as 
the  combinatory  product  of  p  into  the  complement  of  g,  and 
not  as  a  different  kind  of  product  of/  into  q. 

The  term  co-product  may  be  applied  to  such  expressions, 
regarded  as  an  abbreviation  merely,  after  the  analogy  of  cosine 
for  complement  of  the  sine. 

Consider  next  a  unit  normal  vector  system.  By  the  defini- 
tion we  have 

l^=  '2'    \h=   Khi)  =  —  hy 
because       zj  ?,  =  i^i^  =■  i. 


Also,     i 
Next  let 


;;/,/,  -[-  f'l^h     and     e„  =  n^i^  -\-  n^i^ 


*  Gra'^smann  (1S62),  Chapter  4. 


36  grassmann's  space  analysis. 

then,  by  {b)  and  {c), 

I  e,  =  in^  1 1,  +  ;//,  1 1^  =  ;/z,z,  —  m,i^.  (53) 

By  the  figure  it  is  evident  that  |  e,  is  a  vector  of  the  same 
length  as  e^  and  perpendicular  to  it,  or,  in  other  words,  taking 
the  complement  of  a  vector  in  plane  space  rotates  it  positively 
through  90°. 

The  co-product  e,  j  e^  is  the  area  of  the  parallelogram,  two 
of  whose  sides  are  e,  and  [e^  drawn  outwards  from  a  point;  if 
e.  is  parallel  to  |  e^ ,  this  area  vanishes,  or  eje^  =  o;  but,  since 
I  e.,  is  perpendicular  to  e^ ,  e^  must  in  this  case  be  perpendicular 
to  e^ ;  hence  the  equation 

ej  6,  =  o  (54) 

is  the  condition  that  two  vectors  e,  and  e^  shall  be  perpendicu. 
lar  to  each  other. 

The  co-product  e J  e, ,  which  will  usually  be  written  e,-,  and 
called  the  co-square  of  e, ,  is  the  area  of  a  square  each  of 
whose  sides  has  the  length  Te^ ;  hence 

^e,  =yij^=/i^  (55) 

Let  a^  and  a^  be  the  angles  between  i^  and  e,  and  between 
z,  and  62  respectively,  as  in  the  figure.     Then 

e^e,  =  ;;/,«,  —  w,;^,  =  Te,Te„  sin  {a.^  —  a^,  (56) 

the  third  member  being  the  ordinary  expression  for  the  area  of 
the  parallelogram  e,e^.     Also 

=  w,;/,  +  in^iK_  =  Te,  Te,  cos  {a,  —  «',),          (57) 

the   last   member  being   found   as    before,   remembering  that 
sin  (90°  -\-  a^—  «j)  =  cos  {a^  —  a^). 

If  in  (57)  we  let  e„  =  e, ,  whence  ;/,  =  ;;/,  and  n„  =  vi, ,  we 
have 

7e.  =  Ve^  =  Vm:  ^  m^.  (58) 

\i  Te,  =  Te.,  =  i,  then  ;;/,  —  cos  a^,  w.,  =  sin  o',,  «,  =  cos  a„, 
n^  =  sin  a„ ,  and  equations  (56)  and  (57)  give  the  ordinary  trigo- 
nometrical formulas  sin(«a'5  —  a-,)  =  sin  a^  cos  a,  —  cos  cr.,  sin  a^, 


THE    COMPLEMENT. 


37 


and  cos  {a^  —  a-,)  =  cos  o',  cos  6^;,+  sin  a^  sin  a,.     Squaring  and 
adding  (56)  and  (57),  there  results 

re, .  T'e,  =  e.-e,?  =  (e,e,)^  +  (e,  |  €,)\  (59) 

Attention  is  called  to  the  fact,  which  the  student  may  have 
already  noticed,  that  such  an  equation  as  AB  =  AC,  in  which 
AB  and  AC  a.re  combinatory  products,  does  not,  in  general, 
imply  that  B  —  C,  for  the  reason  that  the  equation  A{B—C)=o 
can  usually  be  satisfied  without  either  factor  being  itself  zero. 
Thus  pL,  —  p^i  means  simply  that  the  two  quantities  which 
are  equated  have  the  same  magnitude  and  sign,  which  permits 
L.^  to  have  an  infinity  of  lengths  and  positions,  when  p  and  L, 
are  given.  The  equation  />,/>,  =  A  A »  or/i(A  —  A)  =  O' A  ^"d 
A  being  unit  points,  implies,  however,  that  A  =A'  unless  /^  is 
at  00  ,  that  is,  a  vector. 

Exercise  9. — A  triangle  whose  sides  are  of  constant  length 
moves  so  that  two  of  its  vertices  remain  on  two  fixed  lines  : 
find  the  locus  of  the  other  vertex. 

Let  r„6j  and  r„e„  be  the  two  fixed  lines, 
and  pp' p"  the  triangle.  Let  pe  be  per- 
pendicular to  p' p" ,  /'  —  ^0  =  -^'€1  and 
P"  —  ^0  =  y^i  !  then  /"  —  p'  =  j'e„  —  xe, , 
T{ye.,  —  -re, )  =  r  =  constant,  by  the  con- 
ditions.  Also,  Tp'e  =  constant  =  7/ic, 
say,  and  Tep  =  constant  =  7ic,  say.     Hence 


e  -p'  =  Tp'e.  U{e  -p')  = 


ye,  —  xe^ 


=  in{ye^  —  xe,), 


T{ye,  -  xe,) 
and  similarly/  —  e  =  n\{ys^  —  xe,).     Therefore 

/  —  ^0  =  p  =  ^e,  4-  in{y€,  —  xe,)  +  ti\{ye,  —  xe,), 
an  equation  which,  with  the  condition  T{y€„  —  xe,)  =  c,  or 

y'e,^-  -  2xye,  \  e,  +  x'e,^  =  c\ 
determines  the  locus  to  be  a  second-degree  curve,  which  must 
in   fact  be   an   ellipse,  since  it   can   have  no  points  at  infinity. 
Let  us  rearrange  the  equation  in  p  thus  : 

p  =  x\{  I  —  vi)e,  —  11 1  6,]  -f  j'[we,  +  11 1  e,]  —  xe-\-  ye,  say, 


38 


GRASSMANN  S    SPACE    ANALYSIS. 


SO  that  e  =  {i  —  in)e^  —  n\e^  and  e'  =  me^-\-  n\e^\  then  multi- 
ply successively  into  e  and  e';  therefore  pe  =  ye'e  and 
pe'^xee.  Substituting  these  values  of  a- and  j  in  the  equa- 
tion of  condition,  we  have 

el  ■  {P^y  +  26, 1  €,.pe  .  pe'  +  ejipej  =  r{ee'y, 

a  scalar  equation  of  the  second  degree  in  p. 

Exercise  lO. — There  is  given  an  irregular  polygon  of  ?i 
sides:  show  that  if  forces  act  at  the  middle  points  of  these 
sides,  proportional  to  them  in  magnitude,  and  directed  all  out- 
ward or  else  all  inward,  these  forces  will  be  in  equilibrium. 

Let  r„  be  a  vertex  of  the  pol}-gon,  and  let  2e,,  2e„,.  ..  26„ 
represent  its  sides  in  magnitude  and  direction.  Then  the  mid- 
dle points  will  be  ^„ -j- e, ,  f^-\-  2€,  -}-  6„ ,  etc.,  and,  using  the 
complement  in  a  vector  system,  we  have 

2/>e  =  (^+6,)  1  e,+(r„+2e,+e,)  |  e,+(^„+2e,+2e,+e3)  \e,-\-.... 

+  (^u  +  26,  +  .  . .  +  26„_i  +  e„)  I  e„ 


Ve+>' 6^  +  26, 


^e  -f-  2  6„ 


^^e-f-...4-2e„_i|6„ 


e -f-  (-^  6    =  O,  which  was  to  be  proved. 


Exercise  ii. — A  line  passes  through  a  fixed  point  and  cuts 
two  fixed  lines;  at  the  points  of  inter- 
section perpendiculars  to  the  fixed  lines 
are  erected  ;  find  the  locus  of  the  inter- 
section of  these  perpendiculars. 

Let  the  fixed  lines  be  r„e,  and  c^e.,, 
and  the  fixed  point  (\  -f~  ^s !  the  moving 
line  cuts  the  fixed  lines  in  /'  and  /". 
at  which  points  perpendiculars  are 
erected  meeting  in/. 

Let  p  —  c^  =  p,  p'  —  e„  =  xe,  ,  p"  —  e,  —  je, ,  Te,  =  T^e,  =  I ; 


then  p-=  xe.^x'  \6^=^  y6„-\-y'  \€„  ,  whence  pj  e,=  .r  and  p|  e„=  jj' 


EQUATIONS    OF    CONDITION,    AND    FORMULAS. 


39 


Also,  since  e^  -(-  e^,  p',j>"  are  collinear  points, 

(-*'e,  -   e3)(je,  —  ej  =  o  =  xye^e^^  ye^e^  +  X6^e^\ 
or,  substituting  values  of  x  and  y, 

P\e,.  p\e^.  e,e,  4-p|e,  .e,e3+p|c  .€36,  =0, 
an  equation  of  the  second  degree  in  p,  and  hence  representing 
a  conic. 

Prob.  15.  If  a,  b,  c  are  the  lengths  of  the  sides  of  a  triangle,  prove 
the  formula  a^  =l  b"  -\-  r  —  2bc  cos  A,  by  taking  vectors  e,,  e^,  and 
65  —  ei  equal  to  the  respective  sides. 

Prob.  16.  If  ^„6j  and  e^e-i.  are  two  unit  lines,  show  that  the  vec- 
tor perpendicular  from  e^  on  the  line  {e^  +  ^^i)(^o  +  ^^2)  is 

^^^1^2         , ,,         ■       X      .    ,  .  ,     ,     ,         ,   .          abe^e, 
.   [bei  —  ae),  of  which  the  length  is  -— *-? — -.     From 

this  derive  the  Cartesian  expression  for  the  perpendicular  from  the 
origin  upon  a  straight  line  in  oblique  coordinates, 
ab  ?,\r\  CfO  -4-  {a''  +  b'  —  2ab  cos  go)^,  go  being  angle  between  the  axes. 
Prob.  1 7.- If  three  points,  we^  +  ne^,  me^  -{- ne,,  me.  +  ne^,  be 
taken  on  the  sides  of  the  reference  triangle,  then  the  sides  of  the 
complementary  triangle,  |  {me^  +  ne^),  etc.,  will  be  respectively  paral- 
lel to  the  corresponding  sides  of  the  triangle  formed  by  the  assumed, 
points  {me^  +  Jie^),  {me^  +  "f^^),  etc. 


Art.  9.     Equations  of  Condition,  and  Formulas. 

Several  equations  of  condition  are  placed  here  too-ether  for 
convenient  reference  :  some  have  been  already  given  ;  others 
follow  from  the  results  of  Arts.  7  and  8.     When  we  have 


/■A  =  o, 
the  two  points  coincide  ; 

or  2np  =  O,  I 

the  three  points  are  collinear ; 


e,e,  =  o, 


or 


L,L,  =  o, 
or        «,Z,  -|-  n^L^  =  o, 

the  two  lines  coincide  ; 

L,L,L,  =  o, 

or  'SnL  =  o, 

1 

the  three  lines  are  confluent. 
w,e,  -f  ;/,e,  =  o,  (62) 


(60). 


(61) 


J 


the  two  vectors  are  parallel  (points  at  infinity  coincide); 

e,  I  e.  =  o,  (63) 


40  grassmann's  space  analysis. 


the  two  vectors  are  perpendicular  ; 


either  point  lies  on  the  com- 
plementary line  of  the  other. 


L,\L,  =  o,  (64) 

either  line  passes  through  the 
complementary  point  of  the 
other. 


If  we  write  the  equation 

P  =  -^1^1  +  -^'.e,, 
x^e^  is  the  projection  of  p  on  ei  parallel  to  e,,  and  x^e^  is  the 
projection  of  p  on  e^  parallel  to  e^.     Multiply  both  sides  of  the 
equation  into  e^;    therefore  pe^  =  x^e^e^,  or   x^  =  pe^  -i-  e^e^. 
Similarly,  multiplying  into  e^,  we  have  pe,  =  x^e^e^,  or 
x^  =  pe,  -7-  e„ei,  whence 

p  = + .  (65) 

The  two  terms  of  the  second  member  of  (65)  are  therefore 
the  projections  of  p  on  e,  parallel  to  e^,  and  on  e^  parallel  to  e,, 
respectively.* 

Let  e,  and  e,  be  unit  normal  vectors,  say,  t  and  |z;  then  (65) 

becomes 

p  =  t.  p\t  —  \i .  pi  =  I .  p\i  -\-  ip .  |z;  (66) 

or,  if  Zj  and  i^  be  used  instead  of  i  and  1 1, 

P=  ^^'P\h  +  h-P\h'  (67) 

Again,  in  (65)  let  p  =  e^,  clear  of  fractions,  and  transpose ; 

therefore 

e,e, .  €3+ e.e,.  e,  +  €36,  .e,  =  O,  (68) 

a  symmetrical  relation  between  any  three  directions  in  plane 
space.     Let  T^e,  =  Te^  =  Te^  — •  i,  and  multiply  {6'S)  into  |  €3, 

thus  €,6,  + e^ej.eje,  +  636,  .eje,  =  O,  (69) 

which  is  equivalent  to 

sin  («-  ±  /3)  =  sin  a  cos  yS  ±  cos  a  sin  ft, 

the  upper  or  lower  sign  corresponding  to  the  case  when  €-3  is 

*  Grassmann  (1844),  Chapter  5  (1862),  Art.  129.     Hyde's  Directional  Calcu- 
lus, Arts.  46  and  47. 


EQUATIONS    OF    CONDITION,    AND    FORMULAS. 


41 


L±„ 


between  e^  and  e,,  or  outside,  respectively.  Writing  in  (69)  ]  e^ 
instead  of  e,,  we  have 

e, I  e^  —  ej  63 .  63!  e,  +  €36,  .  e^e^  =  O,  (70) 

which  gives  the  cos  {a  ±  /3).  These  formulas  being  for  any 
three  directions  in  plane  space,  are  independent  of  the  magni- 
tude of  the  angles  involved. 

There  is  given  below  a  set  of  formulas  for  points  and  lines, 
arranged  in  complementary  pairs,  and  all  placed  together  for 
convenient  reference,  the  derivation  of  them  following  after. 

/=(AAA)"'[A  -/AA  +  A  -/AA  +  A  -/AoAl-     ) 
L={L,L,Ly[L, .  LL,L,  +  Z. .  LL^L,  +  4  .  ZZ„ZJi ' 

p={pj.p.y\\p.p.-p\p.  +\p.p.-p\p.^  i  pj.-p\p^, 

P.P.-PJ\  =  -  A  -AAA  +  P.'P.P.P, 

=  P^'P.P.P,—    P.'PlP2p.^ 

31,  L,\M, 
M,  L,\M„_ 

LAM,  L,\M, 

A  1^0  Ak>  a!^» 
/. \g.  PA ^1  /. I Qi 

A  1^0  Aki  PMi 
The  complementary  formula  to  {jj^  is  not  given,  but  may 
l>e  obtained  by  putting  Z's  and  M's  for  /'s  and  ^'s. 

Derivation  of  Equations  (7i)-(77). — Equation  (71).  Write 
p  =  x^p^  -\-  x,p,  -\-  x^p^,  and  multiply  this  equation  by  p,p^ ; 

then  p.p^p  =  x,p,p,p„     or     x,  =  pp,p,  -^ p,p,p,. 

Multiplying  similarly  by  p^p,  and  by  p^p,,  we  find 

^1  =  PP^Po -^  P.P.P.  and  x^=  pp,p,-^p,p.p,.     The  substitu- 


AA-  ki  = 

PMx9^  = 

A  A I  ^i^s  — 


— 

A  AU 
A  Ak 

k7 

1  Al^i 

2  PM-i 

> 

^2  kl     ^2  I  ^2 


L,LAM,= 

L.AM,M,= 
L,LAM,M,= 


P.PlP-X    ■    ^0^.^2    = 


(71) 
(72) 

(73) 

(74) 
(75) 
(76) 

(77) 


43 


GRASSMANN  S   SPACE    ANALYSIS, 


tion  of  these  values  gives  the  first  of  (71),  and  the  second  is 
similarly  obtained  or  may  be  found  by  simply  putting  Z's  for 
/'s  in  the  first. 

Equation  (72).  Write/  =  x^  \p,p,  +  x\  \  p,p,  +  x,  \  p,p^,  and 
multiply  into  \p^ ;  thus/|/„  =  ^\P^Pxpy  Find  in  the  same  way 
values  of  x^  and  x„,  and  substitute. 

Equation  (73).  Write />,/„  .p^p^  ~  xp^  -\- ypi-,  and  multiply 
by//, ;  therefore//,  .p,p,.p,p,  =  xpp.,p,,  or,  by  Eq.  (38), 
P.PP.-P.P^P.  =  -'-PP.Px  =  -  ^P-.PP:^  ox,x=- p,p,p,.  Multiply- 
ing by//,  we  find  y  —  p^p.p,,  and  on  substituting  obtain  the 
first  of  {JT)).  For  the  second  put  /,/,  .p^p^  =  xp^  +  j/4,  and. 
proceed  in  a  similar  way. 

Equation  (74).   In  the  first  of  (73)  put  /^^p^  =  |^,. 

Equation  (75).   In  the  fourth  of  (73)  put 

-^1-^2    ^^^  Pit    ^%^^^     l^l»    ^4    ^    1^2- 

Equation  {^6).   Multiply  (75)  by/,. 

Equation  ij"]).  In  the  first  of  (72)  put  q.,  for/,  and  multiply 
by  A/.  ^.-^0^1  ;  then 

/"o/.A  •  QS.q-^.  =  ^.?,  \P^p2  •  ^.  \Po+  9.9  ^  \P.P.  •  Qi  1/.+  ^0^7, 1  A/*.  •  q^  I A 


Ak. 


p\q.  Al^. 
Ai^o  A!^i 


+Ak. 


+  Ak. 


Ai^o  Al^i 
A  1^0  A,'^> 


Al^n    A'^, 

by  (76),  which  is  equivalent  to  the  third  order  determinant  of 
equation  {77)-^ 

Exercise  12. — To  show  the  product  of  two  determinants  as 
a  determinant  of  the  same  order. 

Let  /„  =  ^le,  /,  =  2vtc,  /,  =  'S7ie,  q^  =  2Xe,  q^  =  2 pie,  q.,  =  2ve; 

then  A/,  A  =  [^0-  '^^'  '^]'  q,q,q.  =  [^c  /^,»  n]  ;  also 
A  ko  =  ^0^0  +  A^,  +  ^.^.'  /,  ko  =  ^'^o'^o  +  '«.^.  +  ^^^X.,  etc.    Sub- 
stituting these  values  in  (77),  we  have  the  required  result.     A 

solution  may  also  be  obtained  directly  without  the  use  of  {77). 

2 

Let  the  ^'s  be  as  above,  but  write  A  =  ^^'-/'i  =  2wq,p.^  =  2nq. 

0 

Then 

p^p^p^  =  2/q.2P!q.2uq=[/„,  m^,  n,']q„q,q,  =  [/„  ;«,,  «,][A„,  ;y„  rj. 
*  Grassmann  (1S62),  Art.  173. 


EQUATIONS    OF    CONDITION,    AND    FORMULAS.  43 

Also  /„  =  /„:SA^  +  i;2}xe  +  l^'2ve 

with  similar  values  for/,  and/^j  which  on  being  substituted  in 
P«P\P%  g^^^  '^'^^  result.  Equation  {jj),  however,  exhibits  the 
product  in  a  very  compact,  symmetrical,  and  easily  remembered 
form.* 

Exercise  13. — Show  that  the  sides/,/^?  AA' A/^i  of  the  tri- 
angle/j/^A  cut  the  corresponding  sides  1/3,  |/j,  j/^  of  the  com- 
plementary triangle  in  three  collinear  points. 

The  three  points  of  intersection  are,  using  (74), 

/A-lA==-A-AlA+A-AlA.AA-lA  =  -A-AlA+A-AiA. 
A/i  •  !A  =  ~  A -/i  IA +/!  •  AIA  '  of  which  the  sum  is  zero, 
showing  that  the  points  are  collinear.  It  may  be  shown  in 
the  same  way  that  the  lines  joining  corresponding  vertices  are 
confluent. 

Exercise  14. — If  the  sides  of  a  triangle  pass  through  three 

fixed  points,  and  two  of  the  vertices 

slide  on  fixed  lines,  find  the  locus  of  /\ 

the  other  vertex.  /      \ 

_^ py         \  „ 

Let  the  fixed  points  and  lines  be       ^^3  7><^\' 

/,,  /„  ^3,  Z,,  U,  and  /,  p',  p"  the     ^,-'"'"/  \V 

vertices   of    the   triangle,   as    in    the         ^'  \ 

figure.     Then  p'p^p"  ^o\  p'    coin-  ^ 

cides    with  pp^.L^    and/''    with  pp„.  L„\    hence    substituting 

(//, .  I^^PX^i- Pip)  —  O,  the  equation  of  the  locus,  which,  being 

of  the  second  degree  in  p,  is  that  of  a  conic. 

Prob.  18.  Show  that  if  the  three  fixed  points  of  the  last  exercise 
are  collinear,  then  the  locus  of  /  breaks  up  into  two  straight  lines. 
Use  equation  (73). 

Prob.  19.  If  the  vertices  of  a  triangle  slide  on  three  fixed  lines, 
and  two  of  the  sides  pass  through  fixed  points,  find  the  envelope  of 
the  other  side.  (This  statement  is  reciprocally  related  to  that  of 
Exercise   14,  that  is,  lines  and   points   are  replaced  by  points  and 

*  These  methods  may  be  applied  to  determinants  of  any  order  by  using  a 
space  of  corresponding  order. 


44  grasSiMANn's  space   analysis. 

lines  respectively,  and  the  resulting  equation  will  be  an  equation  of 
the  second  order  in  Z,  a  variable  line.) 

Prob.  20.  Show  that  if  the  three  fixed  lines  of  Problem  19  are 
confluent,  then  the  envelope  of  L  reduces  to  two  points  and  the  line 
joining  them. 

Art.  10.    Stereometric  Products. 

The  product  of  two  points  in  solid  space  is  the  same  as  in 
plane  space.     See  Art.  7. 

Product  of  Three  Points. — Any  three  points  determine  a 
plane,  and  also,  as  in  Art.  7,  an  area  ;  hence /^/../j  is  a  plane-sect 
or  a  portion  of  the  plane  fixed  by  the  three  points  whose 
area  is  double  that  of  the  triangle  p^p^py  It  may  be  shown,  in 
the  manner  used  in  Art.  7  for  the  sect,  that  no  plane-sect,  not 
in  this  plane,  can  be  equal  to/,/„/'3,  and  that  any  plane-sect  in 
this  plane  having  the  same  area  and  sign  will  be  equal  to/j/^/'j.* 
Of  course /',/'„/'3  is  not  now  scalar. 

Product   of    Four  Points. — Anj'  four    non-coplanar    points 

determine  a  tetrahedron,  sav 

ii^ ^ip^  .  ^ 

y  ""--.^  p^p.p^p^,  and  six  times  the  vol- 

"\^     ume    of    this    tetrahedron    is 

£     \      /  ?^^,'  ^^     taken    for    the    value    of    the 

Jy  "v,    y'  product,  because   this    is   the 

^^*  volume  of  the  parallelepiped 

generated  by  the  product /,/,/3, — i.e.  the  parallelogram/^,/,, — 

when  it  moves  parallel  to  its  initial  position  from/,  to /^.     Let 

P.-P.  =  e,  P,-P.  =  e,  p,-p,  =  e",  then 

/.AAA  =  P.P.P.^"  =  P^P.^'^"  =  A^e'e".  (78) 

3  .!  3  3 

If  /,  =:^ke,  p„  =  '^le,  p^  =  '^me,  /,  =  :^ne,  then 
0  "         0  0  0 

p,p.p,p,  -  '^ke^le'2inc'^ne  =  [/-„,  /,,  w,,,  n^  .  i\e^c^e^ ;     (79) 

from  which   it  appears   that  any  two  quadruple   products   of 

points  differ  from  each  other  only  by  a  scalar  factor,  that  is,  they 

differ  only  in  magnitude,  or  sign,  or  both  ;  hence  such  products 

arc    themselves  scalar.f     If  /.AAA  =  O'    ^^^    volume   of  the 

tetrahedron  vanishes,  so  that  the  four  points  are  coplanar. 

*  Grassmann  (1862),  Art.  255.  \  Grassmann  (1S62),  Art.  263. 


STEREOMETRIC    PRODUCTS.  45 

Product  of  Two  Vectors. — The  two  vectors  determine  an 
area  as  in  Art.  7,  but  they  also  determine  now  a  plane  direc- 
tion, so  that  the  product  e^e^  is  a  plane-vector,  and  is  not  scalar 
as  in  plane  space.  Also,  e^e^  differs  from  p^e.e^  now  just  as  e 
differs  from  pe;  namel}',  e,e^  has  a  definite  area  and  plane 
direction,  that  is,  toward  a  certain  line  at  infinity,  whWo.  p^e^e^  is 
fixed  in  position  by  passing  through/,.  Equation  (ly)  there- 
fore  does  not  hold  in  solid  space. 

Product  of  Three  Vectors. — Three  vectors  determine  a 
parallelepiped  as  in  the  figure  above,  and  ee'e"  is  therefore 
the  volume  of  this  parallelepiped.  Any  other  triple  vector 
product  can  differ  from  this  only  in  magnitude  and  sign.     For 

let  616,63  be  such  a  product,  and  write 

333 
6  =  ,r,e,  -f-  x^e^-\-  x^e^  —  ^xe,  e'  =  ^ ye,  e"  =  '^ze ;  then 

1  1  1 


ee'e"  =  '2xe'2ye'2,ze  = 


x^  x^  x^ 

fi  y.  y. 


6. 6^63,  (80) 


so  that  the  two  products  only  differ  by  the  scalar  determinant 
factor.  Hence  the  product  of  three  vectors  must  be  itself  a 
scalar,  by  Art.  i.  Since,  then,  the  product  of  four  points  has 
precisely  the  same  signification  as  that  of  three  vectors,  we  may 
v/rite 

P.PJJ.  =p,ee'e"  =  ee'e"  =  (A  -  AXA  -  /.)(/4  "  A) 

=  P.P.A  -  /s/^/i  +  /./,/.  -  AAP,-  (81) 

Thus  the  sum  of  the  plane-sects  forming  the  doubles  of  the 
faces  of  a  tetrahedron,  all  taken  positively  in  the  same  sense 
AS  looked  at  from  outside  the  tetrahedron,  is  equal  to  the 
volume  of  the  tetrahedron.      Compare  equation  (37). 

If   ee'e"  =  O,   the   volume   of   the   parallelepiped   vanishes, 
and  the  three  vectors  must  be  parallel  to  one  plane. 

Product  of  Two  Sects. — In  solid  space  two  sects  determine 
a  tetrahedron  of  which  they  are  opposite  edges.     Thus 

AAAA  =/'A.  •  AA  =  ^.A  =  AA  -AA  =  AZ„  (82) 
so  that  the  stereometric  product  of  two  sects  is  commutative, 
and  has  the  same  meaning  as  that  of  four  points. 


46  grassmann's  space  analysis. 

Product  of  a  Sect  and  a  Plane-Sect. — Let  them  be  L  and 
P,  and  let  /„  be  their  common  point;  take  p^,  p^,  p^  so  that 
L—p^p^  and  P  =  p„p.2p^-  L  and  P  evidently  determine  the 
point  /„,  and  also  the  parallelepiped  of  which  one  edge  is  L 
and  one  face  is  P,  so  that  the  product  should  be  made  up  of 
these  two  factors.      Hence  we  write 

LP  =  pj, .  /'„AA  =  Pop  J  J, .  Po'y 


PL  =  pjj, .  A/,  =  AAA/,  -/o  =  LP.  )  ^^^^ 

If  L  is  parallel  to  P,  p^  is  at  infinity,  and,  replacing  it  by  e, 

(83)  becomes 

PL  =  LP=  ep,  .  ep,p^  =  epj,p, .  e.  (84) 

Product  of  Two  Plane-Sects. — Let  them  be  P^  and  P,^,  and 

let  L  be  their  intersection,  while/j  and  A  are  such  points  that 

P,  =  Lp^  and  P^  =  Lp^\    then  P^  and  P^  determine  the  line  L 

and  also  a  parallelepiped  of  which  they  are  two  adjacent  faces, 

and 

P.P.,  =  Lp, .  Lp,  =  Lpj^  .L=-  P,P,.  (85) 

If  Pj  and  P.,  are  parallel,  L  is  at  infinity,  and  is  equivalent 
to  a  plane-vector,  say  to  ?/ ;  hence,  substituting  in  (84), 

PA  =  vp, .  vp.  =  np.P.  • '/  =  -  P.P.-  (86) 

Product  of  Three  Plane-Sects. — By  (85)  and  (83)  this  must 
be  the  square  of  a  volume  times  the  common  point  of  the 
three  planes;  or,  if /„' /i' A:' A  t>e  taken  in  such  manner  that 
P.  =PJ.P.^  P.  =AAA'  P.  ^P.P.P.^  then 
P,P,P,^  021.  oil  .012  =023.0123.01  ={p,p,pJ,Y  .p,\  (87) 
the  suffixes  being  used  instead  of  the  corresponding  points. 
If />„  be  at  infinity,  the  three  planes  are  parallel  to  a  single  line, 
and  may  be  written  P,  =  n^ep^p^,  etc.,  and  then  treated  as 
above. 

Product  of  Four  Pla-ne-Sects.* — Let  the  planes  be  /*„ .  .  .  P^, 

and  let/„  .  .  .  />,  be  the  four  common  points  of  the  planes  taken 

three  by  three.     w„  .  .  .  tt^  may  be  so  taken   that  P^  =  '^o/,AA' 

.£tc. ;  then 

P.P^A,  =  n,n.n.^ii, .  123  .  230 .  301  .  012 

=  n^i'.'iMPoPJJ^.y-  (88) 

*  Grassmann  (1S62),  Art.  300. 


STEREOMETRIC    PRODUCTS. 


47 


Product  of  Two  Plane- Vectors. — Let  //^  and  -q^  be  two  plane- 
vectors  or  lines  at  infinity ;  let  e  be  parallel  to  each  of  them, 
and  €1  and  e^  so  taken  that  ?/,  =  ee^,  i]^  =  ee^,  then 

ViV,  =  e^i  •  ee,  =  ee^e, .  e  =  —  7/^7/,,  (89) 

because  7/^  and  7/^  determine  a  common  direction  e,  and  a  paral. 
lelepiped  of  which  three  conterminous  edges  are  equal  to 
€,  ej,  €2,  respectively. 

Product  of  Three  Plane-Vectors. — Take  €^,  e^,  €3  so  that 

'/.V.V3  =  »  '  e,e,  .  €36.  .€,€,  =  ii{e,e^ej.  (90) 

The  directions  e,  .  .  .  €3  are  common  to  the  plane-vectors 
?/j  .  .  .  //s  taken  two  by  two. 

Several  conditions  are  given  here  together  which  follow 
from  the  results  of  this  article. 


Two  points  coincide. 

Three  points  coUinear. 

AAAA  =/.A-AA 
=  Z^Z,  =  o, 
Four  points  co planar ;  two 
lines  intersect. 
e^e„  =  o, 
Vectors  parallel. 

616263  =  O, 
Three  vectors   parallel   to 
one  plane. 


PA  =  o,  (91) 

Two  planes  coincide. 

P.P.P.  =  o,  (92) 

Three  planes  collinear. 
P  P  P  P  =  P  P  .P  P 

=  L,L,  =  o,         (93) 
Four    planes    confluent ;    two 
lines  intersect. 
V.V,  =  O,  (94) 

Plane-vectors  parallel. 

'AV^^s  =  0,  (95) 

Three  plane-vectors  parallel  to 
one  line. 


Sum  of  Two  Planes. — Let  them  be  P^  and  P^,  let  Z  be  a 
sect  in  their  common  line,  and  take  />,  and/,  so  that  /*,  =:  Z,/,, 
P^  =  Lp^ ;  then 

/^.  +  P,  =  Z(/.  +  A)  =  2Lp,  (96) 

p  being  the  mean  of /^  and  p^.     Also 

z'. -Z'^^AA-A);  (97) 

whence  the  sum  and  difference  are  the  diagonal  plane  through 
Z,  and  a  plane  through  Z  parallel  to  the  diagonal  plane  which 
is  itself  parallel  to  Z,  of  the  parallelepiped  determined  by  P^ 


48  grassmann's  space  analysis. 

and  P^.  If  TP,=  TP^,  P,  ±  P^  will  evidently  be  the  two 
bisecting  planes  of  the  angle  between  them.  The  bisecting 
planes  may  also  be  written 

Yp^Yp     °'    P^TP^^P-^TP,,  (98) 

If  the  two  planes  are  parallel,  let  /;  be  a  plane-vector 
parallel  to  each  of  them,  that  is,  their  common  line  at  infinit)-, 
and  let/,  and  p^  be  points  in  the  respective  planes;  then  we 
may  write  P^  =  n^pj},  P^  =  '^sA^'  whence 

^,  +  A  =  k^hp.  +  ^I'.P^v  =  ('^  +  ''J/v-  (99) 

If  «,  -|-  11^  =  o,  this  becomes 

^:  +  A  =  «.(A-A)'A  (100) 

the  product  of  a  vector  into  a  plane-vector  and  therefore  a 
scalar,  by  (80). 

Two  plane-vectors  may  be  added  similarly,  since  they  will 
have  a  common  direction,  namely,  that  of  the  vector  parallel 
to  both  of  them. 

Exercise  15. — If  two  tetrahedra  e^e^e^e^  and  e^e^e^e^  are  so 
situated  that  the  right  lines  through  the  pairs  of  corresponding 
vertices  all  meet  in  one  point,  then  will  the  corresponding  faces 
cut  each  other  in  four  coplanar  lines. 

The  given  conditions  are  equivalent  to  ^//  .  e^e^  =  o 
■^  e  e  '  e  c  '  ^^  e  e  '  .  e  c  '  ^^  e  c  ' .  c  e  '  ^  e  e  '  .€  c  '  '=■  e  e  '  €  e  ' . 
Two  of  the  intersecting  lines  of  faces  are  ^//^  .  ^„V,V/  and 
£'/„c^ .  ^/r„V/,  and,  if  these  intersect,  we  must  accordingly  have, 
by  (93),  012  .  o'i'2'  .  123  .  i'2'3'  =  o  =  012  .  123  .  o'i'2'  .  i'2'3' 
=  0123 .  o'i'2'3' .  I2i'2',  the  last  factor  of  which  is  equivalent 
to  the  fourth  condition  above,  since  quadruple-point  products 
in  solid  space  are  associative.  Similarly  all  the  other  pairs 
of  intersections  may  be  treated. 

Exercise  16. — The  twelve  bisecting  planes  of  the  diedral 
angles  of  a  tetrahedron  fix  eight  points,  the  centers  of  the 
inscribed  and  escribed  spheres,  through  which  they  pass  six 
by  six. 

The  sum  and  difference  of  two  unit  planes  are  their  two 


STEREOMETRIC    PRODUCTS.  49 

bisecting  planes,  by  (97).  Let  the  tetrahedron  be  e^e^e^e^,  and 
let  the  double  areas  of  its  faces  be  A^  =  Ti\e^e^,  etc. ;  then  a 

pair  of  bisecting  planes  will  be    " ,'  '  ±_  —~  or  e^clA^c\  ±  ^363)- 

The  pair  through  the  opposite  edge  will  be  e^cj^A^e^  ±  ^,^,)' 
If  there  be  a  point  through  which  the  six  internal  bisecting 
planes  pass,  it  must  be  on  the  intersection  of  these  two  planes 

taken  with  the  upper  signs,  and  we  infer  by  symmetry  that  it 

3 
must  be  the  point  '^Ae.     Another  internal   bisecting  plane  is 

0 

e^e^jyA^e^  -\-  A^c^,  which  gives  zero  when  multiplied  into  '^Ae^ 
as  do  also  the  other  three. 

To  obtain  all  the  points  we  have  only  to  use  the  double 
signs,  so  that  they  are  ±  A^e^  ±  A^c^  ±_  A^l\  ±_  A^e^.  This 
gives  eight  cases,  namely, 

+   +   +  +  -+  +  + 

+  +   +   -  +  + 

+   +-+  + + 

+   -++  +-   +   - 

The  eight  apparent  cases  that  would  arise  by  changing  all  the 
signs  are  included  in  these  because  the  points  must  be  essen- 
tially positive.  Moreover,  no  positive  point  could  have  three 
negative  signs,  because  the  sum  of  any  three  faces  of  the  tetra- 
hedron must  be  greater  than  the  fourth  face.  It  will  be  found 
on  trial  that  six  of  the  bisecting  planes  will  pass  through 
2(  ±  Ae)  with  any  one  of  the  above  arrangements  of  sign. 

Prob.  21.  The  twelve  points  in  which  the  edges  of  a  tetrahedron 
are  cut  by  the  bisecting  planes  of  the  opposite  diedral  angles  fix 
eight  planes,  each  of  which  passes  through  six  of  them. 

Prob.  22.  The  centroid  of  the  faces  of  a  tetrahedron  coincides 
with  the  center  of  the  sphere  inscribed  within  the  tetrahedron 
whose  vertices  are  the  centroids  of  the  respective  faces  of  the  first 
tetrahedron. 

Prob.  23.  If  any  plane  be  passed  through  the  middle  points  of 
two  opposite  edges  of  a  tetrahedron,  it  will  divide  the  volume  of  the 
tetrahedron  into  two  equal  parts. 


50  grassmann's  space  analysis. 

Art.  11.     The  Complement  in  Solid  Space. 

According  to  the  definitions  of  Art.  8  the  complementary 
relations  in  a  unit  normal  vector  system  are  as  follows : 

hh=  1(10  =  z,  I 

^3^  =  1(10  =  h  r-  (loi) 


I,  =   Kl. 


'•s'-i ' 
=   hi,, 


^-^^(^^. 


Let  e  =  ^li ;  then 

I  e  =  /,hh  +  ^.'s^,  +  ^.hh  =  j{Ki,  -  /.z.)(/,^  -  /,0.  (I02) 

so  that  |e  is  a  plane-vector.     The  figure,  which  is   drawn    in 

isometric    projection,  shows 
that  the  two  vectors  /./.,  —  /„/j 
i,i;.    Vs*     f^  ^"<^    O3  ~  ^s'l'    whose    prod- 

uct is  /,  •  I  e,  are  both  perpen- 
dicular to  e;  for  the  first  is 
perpendicular  to  /,;,  -^  /„;.,, 
which  is  the  orthogonal  pro- 
jection of  eupon  ZjZ^,and  to 
Zj,  and  therefore  is  also  per- 
pendicular to  e,  while  the 
second  is  perpendicular  to  /^z,  -|-  l^i^  and  to  z.^,  and  therefore 
to  e.  Hence  |  e  is  a  plane-vector  perpendicular  to  e  ;  and,  since 
|(|e)  =  e,  the  converse  is  also  true,  i.e.  the  complement  of  a 
plane-vector  is  a  line-vector  normal  to  it. 

The  figure  shows  that  e  is  equal  to  the  vector  diagonal  of 
the  rectangular  parallelepiped  whose  edges  have  the  lengths 
/,,  /„  /,,  hence 

Te  =  4//7  + 1:  + 1;.  (103) 

Multiply  equation  (102)  by  e;  therefore 

el  6  =  (/,z,  +  Kj„_  +  /^'^V/,^/,  +  Kr^i,  +  /3^0 

=  /;+C  +  /3^=7'V  =  ei,  (104) 

so  that  the  co-square  of  a  vector  is  equal  to  the  square  of  ita 
tensor.  The  product  e|e  is  that  of  a  vector  e  into  a  plane- 
vector  perpendicular  to  it,  as  has  just  been  shown  ;  it  is  there- 


THE    COMPLEMENT    IN    SOLID    SPACE. 


51 


fore  a  volume  which  is  equivalent  to  Te .  T\e\  hence,  by  (104), 
eje=  Te .  T\e  ■=  T^e,  or  7^6=  T\e.  Hence,  the  complement 
of  a  vector  in  solid  space  is  a  plane-vector  perpendicular  to  it 
and  having  the  same  tensor,  or  numerical  measure  of  magni- 
tude.* 

3 
Let  a  second  vector  be  e'  ^^ini ;  then 

1 

6 1  e'  =  /,///j  -\-  /„7/V„  -\-  I^m^  ==  e'  I  e.  (105) 

Now  e\e'    being  the  product  of  e   into  the  plane-vector  ]  e', 
is  the  volume  of  the  parallelepiped  in  the  fig- 
ure, that  is,  TeTe'  sin  (angle  between  e  and  |  e') 
=  ZeT^e'cosf.      Hence 

e\e'  —  e'  \  e=l^in^~\-l^vi„-\-l^i!i^=  TeTe  cos  f.  (106) 

If    Te—  Te'  —  \,  l^  .  .  .  l^,  w,  ,  .  .  111^  are  di- 


rection cosines,  and  (105)  gives  a  proof  of  the 

formula  for  the   cosine  of    the   angle   between 

two  lines  in  terms  of  the  direction  cosines   of  the  lines.     We 

have  also  in  this  case 

ee   =  (/,;//,  -  /„;//,)  1 1^  +  {J^m^  —  l^m^  \  i^  -f  {l^m^  —  l,in^)  \  /„,    and, 

taking  the  co-square, 

{eej=  (sin  :')=  =  (/,;//,-  l,my+{hn-l,vi:)'  +  {l^m-l^my.  (107) 

If  e\e' =0,  (108) 

6  is  parallel  to  the  plane-vector  perpendicular  to   e',  that  is,  e 
is  perpendicular  to  e' ,  as  is  also  shown  by  (106). 

Let  ?/  =  I  e,  if  =  I  e'  ;  then 
r]\jf  =■  1  e  .  e'  =  e'  I  e  =  e|  e'  =  TeTe'  cos  f  =  TfjTif  cos  ^'.  (109) 
and  7;|r/'  =  O  (l  lO) 

is  the  condition  of  perpendicularity  of  two  plane-vectors.     Also 

either 

e|7/  =  o,     or     //'|e  =  o,  (nO 

is  the  condition  that  a  vector  shall  be  perpendicular  to  a  plane- 
vector,  for  the  first  means  that  e  is  parallel  to  a  vector  which  is 


*  Grassmann  (1862),  Art.  335. 


GRASSMANN  S    SPACE    ANALYSIS. 


perpendicular  to  ?/,  and  the  second  that  ?/  is  parallel  to  a  plane- 
vector  which  is  perpendicular  to  e. 

Equations  (7i)-(77)  of  Art.  9  become  stereometric  vector 
formulae  if  e,,  e^,  etc.,  be  substituted  for />,,/,,  etc.,  and  ?;, ,  y^. 
etc.,  for  Zj,  Z,,  etc.     For  instance,  {jG)  gives  the  vector  formulas 


-12  I    1    i 


>     ViV,  I  7i  V, 


(112) 


^/J^/     '/J '7, 

For  lack  of  space   no  treatment   of    the  complement  in   a 
point  system  in  solid  space  is  given. 

Exercise  17. — To   prove   the   formulas   of   spherical   trigo- 
nometry cos  a  =  cos  d  cos  <:  -|-  sin  d  sin  c  cos  A,  and 
sin  a        sin  d  _  sin  c 
sin  A       sin  i?       sin  6" 
Take  three  unit  vectors  e, ,  e^,  e^  parallel  to  the  radii  to  the 
vertices  of  the  spherical  triangle,  then  «=:(angle  bet.  e„  and  e^), 
A—{a.ugle  bet.  e^e^  and  e^e^),  etc.    In  eq.  (112)  put  e^e^  for  e/e„'; 

hence    e^e, |  e^e^  =  sin  d  sin  c  cos  A  =  e-.  e^\e^  —  e^  |  e^ .  ej  €3 

=  cos  a  —  cos  d  cos  c. 
Again, 

^(e^e, .  6,63)  =  7\e,e.^e3 .  e,)  =  Te^e^e,  =  T(e^6^ .  e,e-,)=  T{€,e.^  .  e.e,)', 

or     sin  i?  sin  c  sin  ^i  =  sin  a  sin  <r  sin  B  =  sin  rt'  sin  /?  sin  6", 
whence  we  have  the  second  result  b}-  dividing  by  sin  a  sin  /^sin  c. 

Exercise    18. — Show  that  in  a  spherical  triangle  taken  as 

ive 


in   Exercise   17,  cos   —  =  —  'vr^ -r—r^ ^^>  whence  den 

2  r(f/e,e,+  f/e,e3) 


the  ordinary  value 


sin  s  sin  (s  —  a) 


sin  I?  sin  c 

Expanding,  the  numerator  becomes   i  4-  Ue^eJ  Ue^e^,  and 
the  denominator  |'^2(i  -\-  6"e,eJ  Ue^e^).     Also  there  is  obtained 

Ue  e\Ue  6   =       '  -'^J^- 
dent. 


The   remainder  is  left   to  the  stu- 


i^u-'  ^1^3 


Prob.  24.  If  e^ ,  €„,  €3,  drawn  outward  from  a  point,  are  taken 
as  three  edges  of  a  tetrahedron,  show  that  the  six  planes  perpen- 


ADDITION    OF    SECTS    IN    SOLID    SPACE.  53 

dicular  to  the  edges  at  their  middle  points  all  pass  through  the  end 
of  the  vector  p  =   -Vr(  I  ^.^3  •  ^r+  I  ^36,  .  e/  +  |  e^e, .  e,').     (Sug- 

gestion.  We  must  have  (p  —  ^ej  |  e,  =  o,  with  two  other  similar 
expressions.) 

Prob.  25.  Show  that  e,  |  ee'  and  ee' .  \  e  are  three  mutually  per- 
pendicular vectors,  no  matter  what  the  directions  of  e  and  e' 
may  be. 

Prob.  26.  Let  e,,  e^,  e^  be  taken  as  in  Prob.  24  ;  let  A^  be  the 
area  of  the  face  of  the  tetrahedron  formed  by  joining  the  ends  of 
these  vectors,  and  2A^  =  Te^e^,  etc.;  also  6*,  =  Angle  between  e,e, 
and  6,^3,  etc.:  then  show  that  we  have  the  relation,  analogous  to 
that  of  Prob.  15,  Art.  8, 

^;^=  A^"--^AJ-^A^''—  2A^A^  cos  6,—  2A^A,  cos  6^—  2A^A^  cos  0^. 
If  ^'j  ...  6*3  are  right  angles,  this  becomes  the  space-analog  of  the 
proposition  regarding  the  hypotenuse  and  sides  of  a  right-angled 
triangle.     (Suggestion.   2A,  =  r(e.,  -  ej{e^  -  e,).) 

Prob.  27.  There  are  given  three  non-coplanar  lines  t'_,e, ,  t^,,e^, 
^^63;  planes  cut  these  lines  at  right  angles,  the  sum  of  the  squares  of 
their  distances  from  c„  being  constant.  Show  that  the  locus  of  the 
common  point  of  these  three  planes  is  {p\e^y-\-{p\e^)'-\-{p\e^y  —  c'', 
if  Te,  =  Te.,  =  re,  =  i. 


Art.  12.    Addition  of  Sects  in  Solid  Space. 

Two  lines  in  solid  space  will  not  in  general  intersect,  so  that 
their  sum  will  not  be,  as  in  eq.  (43),  a  definite  line.  For  let 
/}^e^  and/^e^  be  any  two  sects:  then 

A^i  +  Ae,  =  />,e^  +Ae.  +  ^'„(e,  +  ej  —  ^„(e,  +  e,) 
=  ^o(e,  +  e.J  +  (A  -  Oe,  +  (/,  -  ^„)e,; 
that  is,  the  sum  is  a  sect  passing  through  an  arbitrary  point  e^, 
and  a  plane-vector,  the  sum  of  the  two  in  the  equation.  The 
sum  cannot  be  a  single  sect  unless  the  two  are  coplanar ;  for  let 
/>,  =/,  +  A'^i  +  je^  -j-  .ce^,  63  being  a  vector  not  parallel  to  eje^; 
hence       /.e,  +  p,e^  =  />,€,  +  (/,  +  xe^  +  j/e,  +  ^63)6, 

=  P.  (e,  +  e,)  +  xe^  (e,  +  eJ  +  5-636, 
=  (/,  +  ^ej  (e,  +  eJ  +  ^£36, ; 
and  this  cannot  reduce  to  a  single  sect  unless  2  =  0,  that  is,  un- 
less/^e,  and/^e^  are  coplanar.     Since  a  plane-vector  is  a  line  at 


54  grassmann's  space  analysis. 

CO ,  the  sum  of  two  lines  may  always  be  presented  as  the  sum 
of  a  finite  line  and  a  line  at  co . 

If  the  sum  of  any  two  sects  is  equal  to  the  sum  of  any 
other  two,  their  products  will  also  be  equal,  that  is,  the  two 
pairs  will  determine  tetrahedra  of  equal  volumes.  For  let 
L^-\-  L^  —  L^-\-  L^\  then  squaring  we  have  L^L^  =  L^L^ ,  since 
L^L^  =  o,  etc. 

An  infinite  number  of  pairs  of  sects  can  be  found  such  that 
the  sum  of  each  pair  is  equal  to  the  sum  of  any  given  pair;  for 
let  a  given  pair  be  p^e^  -\- p.2^„,  and  take  a  new  pair 

{xj^  +  ^'.A)(".e,  +  ?'.ej  -t-  {y,p,  +  y,p,){v,e,  +  v^e^) 

{x\n^  '^}\t\)p,e„_  +  {xji,  +J',e\)/,6,. 

This  will  be  equal  to  the  given  pair  if  we  have 
'*'i^^i+ jr^.  =  ^',^^2  4- A"^.  =1.  and  x,u^  +y,v,  =  x^n^  ^-y,v,  =  o). 

Since  there  are  eight  arbitrary  quantities  with  onl}'  four 
equations  of  condition,  the  desired  result  can  evidently  be  ac- 
complished in  an  infinite  number  of  ways. 

Let  /,6,  ./^e^  .  .  .  .  pn€„  be  n  sects,  and  let  5"  be  their  sum, 
and  ^„  any  point,  then 

S=2pe  =  e^2e  -  e„2e  +  ^pe  =  e^:Ee  +  ^{p  -  e,)e,  ....  (i  13) 

1 
the  sum  of  a  sect  and  a  plane-vector  as  before. 

If  2{p  —  e\)e  is  parallel  to  ^e  it  may  be  written  as  the  prod- 
uct  of  some  vector  e'  into  2e,  that  is,  e'2e,  when  the  sum  be- 
comes 5  =  t\^6  4"  e'^e  =  (.?,  -{-  e')^e,  a  sect,  because  ^„  -|-  e'  is 
a  point.  In  no  other  case  does  5  reduce  to  a  single  sect.  If 
26  =  O.  5  becomes  a  plane-vector.  Of  the  two  parts  compos- 
ing S,  the  sect  will  be  unchanged  in  magnitude  and  direction  if 
e^  be  moved  to  a  new  position,  while  the  plane-vector  will  in 
general  be  altered.  It  is  proposed  to  show  that  a  point  (/  may 
be  substituted  for  e„  such  that  the  plane-vector  will  be  perpen- 
dicular to  ^'e.     Writing 

5  =  ^^6  -iq  -  e;)2e  -f  2{p  -  e,)e, 
and,  for  brevity,  putting  q  —  c^  =  p,  ^e  =  a,  2{p  —  e^e  =  \/3, 

so  that 

S  =  qa  —  pa-\-\^,  (114) 


ADDITION    OF    SECTS    IN    SOLID    SPACE.  55 

we  must  have  for  perpendicularity,  by  (in), 

{\/3  —  pa)  j  ar  =  o  =  I  /Sa  —  pa .  \  a, 
or  pa  .\a^a.  p\a  —  p  .  a-  ^\/!ia.  i^^S) 

The  second  member  is  obtained  from  the  first  by  substitut- 
ing in  eq.  (74)  p  ior  p^  and  a  for /'^  and  q^,  in  accordance  with  the 
statement  at  the  end  of  Art.  ii.  If  in  (115)  we  make  p|«  =  o,. 
p  will  be  the  vector  from  t\  to  q  taken  perpendicularly  to  a, 

say 

p,  =  \afi-^  a'-=q^  —  e,.  (116) 

Since  a  and  (3  are  known,  the  required  point  has  been 
found.     Multiply  (115)  by  a;  then,  using  (75), 

—  ap  .  a-  =  pa ,  ai  =  <ar  I  l3a  ^\fi.d-  —  \a .  a\^, 

whence,  substituting  in  (114), 

This  may  be  called  the  normal  form  of  5.* 

The  sects  of  this  article  represent  completely  the  geometric 
properties  of  forces,  hence  all  that  has  been  shown  applies 
immediately  to  a  system  of  forces  in  solid  space.  We  have 
only  to  substitute  the  words  force  and  couple  for  sect  and  plane- 
vector.  The  resultant  action  of  any  system  of  forces  is  5, 
called  by  Ball  in  his  Theory  of  Screws  *'  a  wrench."  The  con- 
dition for  equilibrium  is  S  =  o,  which  gives  at  once 

^e=iO     and     ^(^p  —  e^)e  =  o;  (118) 

since  otherwise  we  must  have  e^^e  =  —  2{p  —  e^)e,  which  is 
an  impossibility.  The  line  q^e  is  the  central  axis  of  the  sys- 
tem of  forces  5. 

Lack  of  space  forbids  a  further  development  of  the  subject, 
but  what  has  been  given  in  this  article  will  indicate  the  perfect 
adaptability  of  this  method  to  the  requirements  of  mechanics. 

Exercise  19. — Reduce  /^e,  -j-  />„e„  =  5  to  its  normal  form. 
-S"  =  ^0(^1  +  ej  +  (/,  —  r„)e,  +  (A  -  ^o)^2-  For  convenience 
suppose/,  and /„  to  be  taken  at  the  ends  of  the  common  per- 

*Grassmann  (1862),  Art.  346. 


56  grassmann's  space  analysis. 

pendicular  on /,e,  and/^e^,  and   moreover  let  ^o  =  il/i  H-Zs). 
/,  —  ^j  =  i  =  —  (/j  —  e^;  then  z|e,  =  i|e,  =  o.     Accordingly 

>^^  ^  ^Xe.  +  e^)  +z(6.  -  ej  =  ^(e.+0+  ^''+'-_|;^y  "'^ .  |(6.  +  6,) 


Hence  the  normal  form  of  5  is 

Exercise  20. — Forces  are  represented  by  the  six  edges  of  a 
tetrahedron  c^e^,  e^c.^,  c^e^,  e^e^,  e^e^,  e^e^\  find  the  5,  reduce  to 
normal  form,  and  consider  the  special  case  when  three  diedral 
angles  are  right  angles.  5  =  ej^e^  +  ^,  +  ^3)  +  ^2^3  +  ^3^1  +  ^^e„ 
=  ^o(e,+ea+e3)+(^2-^.)(^3-^>)=  ^o(e,+e,+e3)+'(e,-e,Xe3-e,) 
=  ^o(e,  +  e,  +  63)  +  6,63  +  €36,  +  e,e, ,  in  which  e,  =  c\  -  e, , 
etc.  Hence 
o-  (.  _!_ {e,e,  +  636.  4-  e,e,) | (e.  +  e^  +  e,)^      ,    ^     ,    ^  n 

^  -  ^^0  +  (iH^-^:T^37  ^^ '  +  ^  +  ^^ 

For  the  rectangular  tetrahedron  let  e^  =:  ai^  ,  e„  =  ^f, , 
63  =  CZ3 ,  /, ,  z^,  Zg  being  unit  normal  vectors.     Then  we  find 

+  ^+7^17-!  (^^■  +  '^'=  +  ^'3). 

Exercise  21. — A  pole  50  feet  high  stands  on  the  ground  and 
is  held  erect  by  three  guy-ropes  symmetrically  arranged  about 
it,  attached  to  its  top  and  to  pegs  in  the  ground  50  feet  from 
the  pole.  The  wind  blows  against  the  pole  with  a  pressure  of 
50  pounds  in  the  direction  e^  —  p,  when  e^  is  at  the  bottom  of 


ADDITION    OF    SECTS    IN    SOLID    SPACE.'  5!? 

the  pole,  and  /  divides  the  distance  between  two  of  the  pegs 
in  the  ratio  —  :  find  the  tension  on  the  guys  and  the  pressure 

on  the  ground. 

Evidently  only  two  of  the  guys  will  be  in  tension  ;  let  their 

pegs  be  at  <?,  and  e^,  and  let  e^  be  at  the  top  of  the  pole,  and  w 

Die  —I—  7ie 

the  weight  of  the  pole.     Then/  =  — ^- ',  and  the  equation 

in  -\-  n  ^ 

of  equilibrium  is 

_    {e,^e^{e,-p)  .  2^elp-e^      (x-\~w}e,e,      ye,e,      ze^e, 

Te^e,  =  50,  Tc^e,  =  Te^e,  =  50  V2,  T{p  -  r„)  =  r(^^^^l±^=  -e) 

{m{e-e:)-^n{e-e:)\  _      ^o  ,        .    -r 

=  T[ 1 ■ 1 — T{me^-\-n£,),  if  e,  =  [/U—e,) 

and  £„  =  f/(^„  —  e,);  then  T{p  —  e,)  = , —  Vm"  +  ;/'  —  mn, 

^  m  -\-  n  ' 

because  e,-  =  e,^  =  i,  and  e^\e^  =  cos  120°  =  —  |,     Hence  the 
equation  of  equilibrium  becomes 

2S^„_{{m  ^  7i)e,  -  me,  -  ne,)  y  z 

.    ,    ,      .       = h  {x  +  tv)e,e^  J^--^e^e,-\ — -^e,e,=o. 

V  in  -\-  n  —  inn  V2  V  2 

Multiply  successively  by  e^e,,  e^e^,  and  e,e,,  and  we  obtain 

,r  +  w_>'_^__  25 

in  -\-  n       in  V2       n  V2        Vi/i^  -\-  ;/'  —  inn 

y  and  z  being  the  tensions,  and  x  -\-  w  the  upward  pressure. 

Prob.  28.  Three  equal  poles  are  set  up  so  as  to  form  a  tripod, 

and  are  mutually  perpendicular;  a  weight  7C'  hangs   upon    a   rope 

which  passes  over  a  pulley  at  the  top   of   the   tripod,  and    thence 

3 
down  under  a  pulley  at  the   ground  at  a  point /  =  ^le,  in  which 

1 

^, .  .  .  ^3  are  at  the  feet  of  the  poles,  and  -^7  =  i ;  if  the  rope  is  pulled 


58  grassmann's  space  analysis. 

so  as  to  raise  w,  show  that  the  pressures  on  the  poles,  supposing  the 
pulleys  frictionless,  are 

Prob.  29.  Six  equal  forces  act  along  six  successive  edges  of  a 
cube  which  do  not  meet  a  given  diagonal;  show  that  if  the  edges  of 
the  cube  be  parallel  to  i,,  i„,  i^,  and  F  be  the  magnitude  of  each 
force,  then  .S  ==  —  2F\  (i,  +  ^^  +  ^s)*  i^  the  diagonal  taken  be  parallel 
to  i,  +  h-\-  h- 

Prob.  30.  Three  forces  whose  magnitudes  are  i,  2,  and  3  act 
along  three  successive  non-coplanar  edges  of  a  cube;  show  that  the 
normal  form  of  ^  is 

-^  =   (^0  +  H^  +  i^.  -  tVO(^   +  2l._+  3h)+  AK^,  +  2l,  +  31,). 

Prob.  31.  Forces  act  at  the  centroids  of  the  faces  of  a  tetrahedron, 
perpendicular  and  proportional  to  the  faces  on  which  they  act,  and 
all  directed  inwards,  or  else  all  outwards;  show  that  they  are  in 
equilibrium. 


INDEX. 


Addition  of  points,  page  9. 

of  weighted  points,  12-16. 
of  vectors,  12. 
of  sects  (planimetric),  30. 
of  sects  (stereometric),  53. 
of  plane-sects,  47. 

Coincidence  of  two  points,  17,  39,  47. 
Collinearity  of  three  points,  17,  39,  47. 
Coplanarity  of  four  points,  17,  39,  47. 
CoUinearity  of  ends  of  three  vectors,  18. 
Coplanarity  of  ends  of  four  vectors,  18. 
Combinatory  products,  24. 

multiplication,  Laws  of,  26. 
Complement,  ^7,,  50. 
Condition,  equations  of,  17,  39,  47. 
Co-product,  35. 
Co-square,  36. 

Difference  of  points,  10. 

Difference  between  p^p^  and  p2~P\,  24. 

Determinants,  product,  42. 

Equations  of  condition,  17,  39,  47. 

Formulae  for  points  and  lines  in  plain 
space,  41. 
for  vectors  in  solid  space,  52. 

Geometric  multiplication,  24. 

Inner  product,  35. 

Inscribed  and  escribed  spheres,  centers. 


Laws  of  combinatory  multiplicdtion,  26. 

Mean  point,  13. 
Multiplication,  Geometric,  24. 

Combinatory,  24. 


Parallelism  of  vectors,  47. 

of  plane-vectors,  47. 
Perpendicularity,  Condition  of,  36,  51. 
Plane  sects.  Product,  46. 

vectors.  Product,  47. 
parallel,  47. 
three  parallel  to  one  line, 

47- 
Planimetric  products,  25,  26. 
Point  at  infinity=  vector,  9. 
Product,  Combinatory,  24. 
Products  of  two  points,  26. 

of  three  points,  27. 

of  two  vectors,  28. 

of  two  sects,  28. 

of  three  sects,  29. 

of  two  determinants,  42. 

Reference  systems,  20. 

Scalar,  definition,  9. 

Sect,  definition,  8. 

Sects,  products,  28,  29,  45. 

products  of  parallel,  30. 

planimetric  sum,  31. 

stereometric  sum,  53. 
Spheres,  inscribed  and  escribed,  centers 

of,  48. 
Spherical  trigonometry,  formulae,  52. 

Tcnsor=  T,  9. 

Unit=  U,  9. 

Vector,  Definition  of,  10. 

plane,  45. 
Vector  products  and  conditions,  17,  18, 

47- 
Wrench,  55. 


SHORT-TITLE     CATALOGUE 

OP  THE 

PUBLICATIONS 

OP 

JOHN   WILEY   &    SONS, 

New  York. 
London:  CHAPMAN  &  HALL,  Limited. 


ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
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AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding ismo,  Si  75 

Principles  of  Animal  Nutrition 8vo,  4  00 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement i2mo,  1  50 

Part  II.  Systematic  Pomology i2mo,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  00 

Elliott's  Engineering  for  Land  Drainage lamo,  i  50 

Practical  Farm  Drainage i2mo,  i  00 

Green's  Principles  of  American  Forestry i2mo,  i  50 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (WoU.) i2mo,  2  00 

Kemp's  Landscape  Gardening i2mo,  2  50 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo,  i  50 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo,  i  50 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Bashore's  Sanitation  of  a  Country  House i2mo,  1  00 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  00 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  00 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  00 

Brigg's  Modern  American  School  Buildings 8vo,  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  00 

Freitag's,  Architectural  Engineering 8vo,  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

1 


Gerhard's  Guide  to  Sanitary  House-inspection lOmo,  i  oo 

Theatre  Fires  and  Panics i2mo,  i  50 

♦Greene's  Structural  Mechanics 8vo,  2  50 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,  ,-5 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  00 

Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,  mor.,  5  00 

Merrill's  Stones  for  Building  and  Decoration Svo,  5  00 

Non-metallic  Minerals:   Their  Occurrence  and  Uses Svo,  4  00 

Monckton's  Stair-building 4to,  4  00 

Patton's  Practical  Treatise  on  Foundations Svo,  5  00 

Peabody's  Naval  Architecture Svo,  7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  00 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo,  i  50 

Snow's  Principal  Species  of  Wood Svo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo,  2  00 

Towne's  Locks  and  Builders'  Hardware iSmo,  morocco,  3  00 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  00 

Sheep,  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo,  5  00 

Sheep,  5  50 

Law  of  Contracts Svo,  3  00 

Wood's  Rustless  Coatings :   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .Svo,  4  00 
Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,  I  25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i  00 


ARMY  AND   NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff' s  Text-book  Ordnance  and  Gunnery Svo,  6  00 

Chase's  Screw  Propellers  and  Marine  Propulsion Svo,  3  00 

Cloke's  Gunner's  Examiner Svo,  i  50 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo.  3  00 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Mihtary  Law  of  United  States Svo,  7  00 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  00 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  00 

Durand's  Resistance  and  Propulsion  of  Ships Svo,  5  00 

*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  00 

Eissler's  Modern  High  Explosives Svo,  4  00 

*  Fiebcger's  Text-book  on  Field  Fortification Small  8vo,  2  00 

Hamilton's  The  Gunner's  Catechism iSmo,  i  00 

*  Hoff's  Elementary  Naval  Tactics Svo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire Svo,  4  00 

*  Ballistic  Tables Svo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .Svo,  each,  6  00 

*  Mahan's  Permanent  Fortifications.    (Mercur.) Svo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i   50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  00 

*  Elements  of  the  Art  of  War Svo,  4  00 

3 


Metcalf' s  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .8vo,  5  00 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  00 

Murray's  Infantry  Drill  Regulations i8mo,  paper,  10 

Nixon's  Adjutants'  Manual 24mo,  I  00 

Peabody's  Naval  Architecture 8vo,  7  50 

*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner i2mo,  4  00 

Starpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

*  Y.'heeler's  Siege  Operations  and  Military  Mining 8vo,  2  00 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Wcodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

Young's  Simple  Elements  of  Navigation 261UO,  mofoc^s-  ?  00 


ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  00 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  00 

Low's  Technical  Methods  of  Ore  Analysis 8vo,  3  00 

Miller's  Manual  of  Assaying i2mo,  1  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.      (Waldo.) i2mo,  2  50 

O'Driscoli's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  :,  00 

Ri»ketts  and  Miller's  Notes  on  Assaying Svo,  3  00 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 


ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy Svo,  4  00 

Gore's  Elements  of  Geodesy Svo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy Svo,  3  00 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy Svo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy Svo,  3  00 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  00 


BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  r  25 

Thomd  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) Svo,  2  00 


CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  i  25 

Allen's  Tables  for  Iron  Analysis Svo,  3  00 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  Svo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Browning's  Introduction  to  the  Rarer  Elenjents Svo,  i  50 

3 


Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,  4  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Eoltwccd.).  .8vo,  3  co 

Cohn's  Indicators  and  Test-papers lamo,  2  00 

Tests  and  Reagents 8vo,  3  00 

Crafts's  Short  Course  in  Oualitative  Chemical  Analysis.   (Schaeffer.).  .  .i2mo,  i  50 
Dolezalek's   Theory  of   the   Lead  Accumulator   (Storage   Battery).        (Von 

Ende.) i2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Effront's  Enzymes  and  their  Applications.     (Prescott.) Svo,  3  00 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  1   25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses i2mo,  2  00 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  00 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  8vo,  3  00 
System  of    Instruction    in    Quantitative    Chemical   Analysis.      (Cohn.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health i2mo,  i   50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  00 

*  Getman's  Exercises  in  Physical  Chemistry i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i   25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  00 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.      (Morgan.) i2mo,  1   50 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hind's  Inorganic  Chemistry 8vo,  3  00 

*  Laboratory  Manual  for  Students i2mo,  i  00 

Holleman's  Text-book  of  Inorganic  Chemistry.      (Cooper.) 8vo,  2   50 

Text-book  of  Organic  Chemistry.      (Walker  and  Mott.) 8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  00 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  00 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 

Keep's  Cast  Iron 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis. i2mo,  i  00 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  00 

*  Langworthy   and   Austen.        The   Occurrence   of  Aluminium  in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  00 

Lassar-Cohn's  Practical  Urinary  Analysis.      (Lorenz.) i2mo,  1   00 

'Application  of  Some   General  Reactions   to   Investigations  in  Organic 

Chemistry.      (Tingle.) i2mo,  1   00 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

L()b's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,  3  00 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  .8vo,  3  00 

Low's  Technical  Method  of  Ore  Analysis 8vo,  3  00 

Lunge's  Techno-chemical  Analysis.      ( Cohn. ) 1 2mo,  i   00 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i   50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe.  .  i2mo,  60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  00 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i   25 

Matthew's  The  Textile  Fibres 8vo,  3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .i2mo,  i  00 

Miller's  Manual  of  Assaying i2mo,  i  00 

Minefs  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) ....  i2mo,  2   50 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,  i   50 

Morgan's  Elements  of  Physical  Chemistry i2mo,  3  00 

*  Physical  Chemistry  for  Electrical  Engineers i2mo,  1   50 

4 


Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  00 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  00 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) izmo,  i  50 

"                    "               "           "             Part  Two.     (TurnbuU.) i2mo,  2  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper,  50 

Pictet's  The  Alkaloida  and  their  Chemical  Constitution.     (Biddle.) Svo,  5  00 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Poole's  Calorific  Power  of  Fuels. Svo,  3  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Reisig's  Guide  to  Piece-dyeing Svo,  25  00 

Richards  and   Woodman's   Air,   Water,  and    Food   from  a  Sanitary  Stand- 
point   Svo,  2  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  00 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  00 

*  Richards  and  Williams's  The  Dietary  Computer Svo,  i   50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) Svo,  raorocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying Svo,  3  00 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo,  3  50 

Disinfection  and  the  Preservation  of  Food Svo,  4  00 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory Svo,  i  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) lamo,  i  00 

Ruddiman's  Incompatibilities  in  Prescriptions Svo,  2  00 

*  Whys  in  Pharmacy i2mo,  i   00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  00 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) Svo,  2  50 

Schimpf 's  Text-book  of  Volumetric  Analysis l2mo,  2  50 

Essentials  of  Volumetric  Analysis i2mo,  i  25 

*  Qualitative  Chemical  Analysis Svo,  i   25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Stockbridge's  Rocks  and  Soils Svo,  2  50 

*  TiUman's  Elementary  Lessons  in  Heat Svo,  i  50 

*  Descriptive  General  Chemistry Svo,  3  00 

Treadwell's  Qualitative  Analysis.     (Hall.) Svo,  3  00 

Quantitative  Analysis.     (Hall.) Svo,  4  00 

Turneaure  and  Russell's  Public  Water-supplies Svo,  5  00 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  1  50 

*  Walke's  Lectures  on  Explosives Svo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  Svo,  cloth,  4  00 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo,  2  00 

Wassermann's  Immune  Sera:  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   i2mo,  I  00 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis Svo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i   50 

Text-book  of  Chemical  Arithmetic i2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

Wilson's  Cyanide  Processes i2mo,  1  50 

Chlorination  Process i2mo,  i  50 

Winton's  Microscopy  of  Vegetable  Foods Svo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo,  2  00 

5 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS    OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments lamo,  3  00 

Bixby's  Graphical  Computing  Table Paper  igi  X  24i  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  00 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  00 

♦Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  00 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  00 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  I  /es's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy. 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  00 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  00 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.).  i2mo,  2  00 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (V/ood.). 8vo,  5  00 

*  Descriptive  Geometry Svo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy '^vo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  moro^-  ~  00 

Nugent's  Plane  Surveying Svo,  3  ^.^ 

Ogden's  Sewer  Design i2mo,  2  00 

Patton's  Treatise  on  Civil  Engineering Svo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry. Svo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo,  5  00 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  00 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  00 

Sheep,  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  00 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  00 

Warren's  Stereotomy — Problems  in  Stone-cutting Svo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i   25 

Wilson's  Topographic  Surveying Svo,  3  50 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo, 

*       Thames  River  Bridge 4to,  paper, 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 
Suspension  Bridges 8vo, 

6 


Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  00 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  00 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  00 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  00 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses Svo,  I   25 

Bridge  Trusses Svo',  2  50 

Arches  in  Wood,  Iron,  and  Stone Svo,  2  50 

Howe's  Treatise  on  Arches Svo,  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel Svo,  2  00 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  00 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses Svo,  2  50 

Part  II.     Graphic  Statics Svo,  2  50 

Part  III.     Bridge  Design Svo,  2  50 

Part  IV.     Higher  Structures Svo,  2  50 

Morison's  Memphis  Bridge 4to,  10  00 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6ino,  morocco,  2  00 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume Svo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) Svo,  2  00 

Bovey's  Treatise  on  Hydraulics Svo,  s  00 

Church's  Mechanics  of  Engineering Svo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Hydraulic  Motors Svo,  2  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Folwell's  Water-supply  Engineering Svo,  4  00 

Frizell's  Water-power Svo,  5  00 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) Svo,  4  00 

Hazen's  Filtration  of  Public  Water-supply Svo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Water-works Svo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits Svo,  2  00 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

Svo,  4  00 

Merriman's  Treatise  on  Hydraulics Svo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics Svo,  4  00 

Schuyler's   Reservoirs   for   Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  Svo,  5  00 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post,  44c.  additional.). 4to,  6  00 

Turneaure  and  Russell's  Public  Water-supplies Svo,  5  00 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  00 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  00 

Williams  and  Hazen's  Hydraulic  Tables Svo,  i  50 

Wilson's  Irrigation  Engineering Small  Svo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 

Wood's  Turbines Svo,  2  50 

Elements  of  Analytical  Mechanics Svo,  3  00 

7 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo, 

Roads  and  Pavements 8vo, 

Black's  United  States  Public  Works Oblong  4to, 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo, 

Byrne's  Highway  Construction 8vo, 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo, 

Church's  Mechanics  of  Engineering 8vo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to, 

♦Eckel's  Cements,  Limes,  and  Plasters Svo, 

Johnson's  Materials  of  Construction Large  Svo, 

Fowler's  Ordinary  Foundations Svo, 

*  Greene's  Structural  Mechanics Svo, 

Keep's  Cast  Iron Svo, 

Lanza's  Applied  Mechanics .  .Svo, 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols Svo, 

Maurer's  Technical  Mechanics Svo, 

Merrill's  Stones  for  Building  and  Decoration Svo, 

Merriman's  Mechanics  of  Materials Svo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  Manual  for  Steel-users lamo, 

Patton's  Practical  Treatise  on  Foundations Svo, 

Richardson's  Modern  Asphalt  Pavements Svo, 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor., 

Rockwell's  Roads  and  Pavements  in  France i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Smith's  Materials  of  Machines i2mo. 

Snow's  Principal  Species  of  Wood Svo, 

Spalding's  Hydraulic  Cement i2mo,    2  00 

Text-book  on  Roads  and  Pavements i2mo,    2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo,    s  00 

Thurston's  Materials  of  Engineering.     3  Parts Svo,    8  00 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy Svo,    2  00 

Part  II.     Iron  and  Steel Svo,    3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,    2  50 

Thurston's  Text-book  of  the  Materials  of  Construction Svo,    5  00 

Tillson's  Street  Pavements  and  Paving  Materials Svo,    4  00 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .i6mo,  mor.,    2  00 

Specifications  for  Steel  Bridges i2mo,     1   25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  ef  Timber Svo,     2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics Svo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel Svo,    4  00 


.  RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  I  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  I  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  1  50 

Railway  and  Other  Earthwork  Tables Svo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  00 

8 


5 

00 

5 

00 

5 

00 

7 

50 

7 

50 

5 

00 

3 

00 

6 

00 

7 

50 

6 

00 

6 

00 

3 

50 

2 

50 

2 

50 

7 

50 

7 

50 

4 

00 

5 

00 

5 

00 

I 

00 

2 

00 

5 

00 

3 

00 

4 

00 

I 

25 

3 

00 

I 

00 

3 

50 

Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  00 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explores'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i   50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  I  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  00 

Philbrick's  Field  Manual  for  Engineers l6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Railroad  Spiral i6mo,  morocco,  1  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  1  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

izmo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  s  00 


DRAWING. 

Barr's  Kinematics  of  Machinery.  . .  .• 8vo, 

*  Bartlett's  Mechanical  Draviring , 8vo, 

*  "  "  "        Abridged  Ed 8vo, 

Coolidge's  Manual  of  Drawing 8vo,  paper 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to, 

Durley's  Kinematics  of  Machines 8vo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo, 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo, 

Jamison's  Elements  of  Mechanical  Drawing 8vo, 

Advanced  Mechanical  Drawing 8vo, 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo, 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo, 

MacCord's  Elements  of  Descriptive  Geometry 8vo, 

Kinematics;   or.  Practical  Mechanism 8vo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams ". 8vo, 

MacLeod's  Descriptive  Geometry Small  8vo, 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo, 

Industrial  Drawing.     (Thompson.) 8vo, 

Moyer's  Descriptive  Geometry 8vo,  2  00 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  00 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  co 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.     (McMillan.) Svo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  i  00 

Drafting  Instruments  and  Operations i2mo,  1   25 

Manual  of  Elementary  Projection  Drawing i2mo,  i  50 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  i  00 

Plane  Problems  in  Elementary  Geometry i2mo,  i   25 

9 


3 

00 

3 

00 

5 

00 

4 

00 

I 

50 

I 

50 

I 

50 

3 

50 

Warren's  Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  5» 

General  Problems  of  Shades  and  Shadows 8vo,  3  00 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50- 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry Svo,  2  50 

Weisbach's     Kinematics    and    Power    of    Transmission.        (Hermann    and 

Klein.) Svo,  5  Oq, 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving lamo,  2  oo- 

Wilson's  (H.  M.)  Topographic  Surveying Svo,  3  50- 

Wilson's  (V.  T. )  Free-hand  Perspective Svo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering Svo,  I  oo- 

Woolf' s  Elementary  Course  in  Descriptive  Geometry Large  Svo,  3  00 

ELECTRICITY  AND   PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  Svo,  3  00 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo,  i   00 

Benjamin's  History  of  Electricity Svo,  3  00 

Voltaic  Cell Svo,  3  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  00 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo,  3  00 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  00 
Dolezalek's    Theory    of    the    Lead    Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Bi*gess.) Svo,  4  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Gilbert's  De  Magnete.     (Mottelay.) .Svo,  2  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  5a 

Holman's  Precision  of  Measurements Svo,  2  00 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  ..  .Large  Svo,  75 

Xinzbrunner's  Testing  of  Continuous-current  Machines Svo,  2  00 

Landauer's  Spectrum  Analysis.     (Tingle.).  . ' Svo,  3  00 

Le  Chateliers  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  00 

Lr.b's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) Svo,  3  00 

*  Lyons'j  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  Svo,  each,  6  00. 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light Svo,  4  00 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .Svo,  i  50. 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I Svo,  2  50 

Thurston's  Stationary  Steam-engines Svo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat Svo,  i   50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo,  2  00 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 

LAW. 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo,  7  00 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  00 

Sheep,  6  so 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo  5  00 

Sheep,  5  50 

Law  of  Contrasts Svo,  3  00 

Winthrop's  Abridgment  of  Military  Law I2m0i  2  50 

10 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Holland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  .the 

Practice  of  Moulding lamo,  3  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  i  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  00 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  SteeL     A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  00 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  4  00 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Rustless  Coatings :   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,  4  00 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

*  Bass's  Elements  of  Differential  Calculus i2mo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo,  i  00 

Compton's  Manual  of  Logarithmic  Computations i2mo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

*  Dickson's  College  Algebra Large  i2mo,  I  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  I  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  5a 

Halsted's  Elements  of  Geometry 8vo,  i  75 

Elementary  Synthetic  Geometry 8vo,  i  50 

Rational  Geometry i2mo,  i  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,  15 

100  copies  for  5  00 

*  Mounted  on  heavy  cardboard,  8X10  inches,  25 

10  copies  for  2  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus .  .  Small  8vo,  3  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,  i  50 

11 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,     3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  00 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.).  i2mo,    2  00 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  00 

Trigonometry  and  Tables  published  separately Each,    2  00 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     i  00 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     1  00 

No.  1.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  Wilham  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
byl  Mansfield  Merriman.  No.  11.  Functioas  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  00 

Merriman  and  Woodward's  Higher  Mathematics 8vo,    3  00 

Merriman's  Method  of  Least  Squares Svo,    2  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Difierential  Calculus. .  Sm.  Svo,    3  00 

Differential  and  Integral  Calculus.     2  vols,  in  one Small  Svo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry Svo,    2  00 

Trigonometry:   Analytical,  Plane,  and  Spherical i2mo,     i  00 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i   50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery Svo,  2  50 

♦  Bartlett's  Mechanical  Drawing Svo,  3  00 

*  "  "  "        Abridged  Ed Svo,  1   50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  00 

Carpenter's  Experimental  Engineering Svo,  6  00 

Heating  and  Ventilating  Buildings Svo,  4  00 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  Svo,  4  00 

Coolidge's  Manual  of  Drawing Svo,  paper,  1  00 

Coelidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i   50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines Svo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power 12 mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i   25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

12 


Button's  The  Gas  Engine 8vo,  5  00 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  I  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo.  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.)  .  .8vo,  4  00 

MacCord's  Kinematics;   or.  Practical  Mechanism 8vo,  5  00 

Mechanical  Drawing 4to,  4  00 

Velocity  Diagrams 8vo,  1  50 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  SO 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design, 8vo,  3  00 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  00 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Thurston's   Treatise   on    Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,  3  GO 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  1 2mo,  i  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — • 

Klein.) 8vo,  5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Turbines 8vo,  2  50 


MATERIALS  OP  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures .  .8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merriroan's  Mechanics  of  Materials 8vo,  5  00 

Strength  of  Materials i2mo,  i  00 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  00 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction 8vo,  5  00 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  00 

13 


Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  00 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram lamo,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .      i6mo,  mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  00 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Heat  and  Heat-engines 8vo,  5  00 

Kent's  Steam  boiler  Economy 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors    8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  00 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  00 

Spangler's  Valve-gears Svo,  2  50 

Notes  on  Thermodynamics i2mo,  1  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Handy  Tables Svo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  Svo,  10  00 

Part  I.     History,  Structure,  and  Theory Svo,  6  00 

Part  II.     Design,  Construction,  and  Operation Svo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake Svo,  5  00 

Stationary  Steam-engines Svo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation Svo,  5  00 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) Svo,  5  00 

Whitham's  Steam-engine  Design Svo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) l6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .Svo,  4  00 


MECHANICS   AND   MACHINERY. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures   Svo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering Svo,  6  00 

Notes  and  Examples  in  Mechanics Svo,  2  00 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo.  i  50 

14 


Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  "   50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's   Record  of   the  Transportation  Exhibits  Building  of  the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  00 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  1  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

,          Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  1  00 

Holly's  Art  of  Saw  Filing iSmo,  75 

James's  Kineruiitics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  00 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery Svo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo,  3  00 

Kerr's  Power  and  Power  Transmission Svo,  2  00 

Lanza's  Applied  Mechanics Svo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods Svo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.). Svo,  4  00 
MacCord's  Kinematics;  or.  Practical  Mechanism Svo,  5  00 

Velocity  Diagrams Svo,  i  50 

Maurer's  Technical  Mechanics Svo,  4  00 

Merriman's  Mechanics  of  Materials Svo,  5  00 

*  Elements  of  Mechanics i2mo,  i  00 

*  Michie's  Elements  of  Analytical  Mechanics Svo,  4  00 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  I Svo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  00 

Smith's  (O.)  Press-working  of  Metals Svo,  3  00 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  00 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo,  3  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work Svo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  I  00 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.).  Svo,  5  00 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). Svo,  5  00 

Wood's  Elements  of  Analytical  Mechanics Svo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines Svo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

15 


METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

**  lias's  Lead-smelting.     (Postage  p  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe Svo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)i2mc.  3  00 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.)...  .i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts Svo,  8  00 

Part    II.     Iron  and  Steel Svo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 


MINERALOGY. 


Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  00 

Map  of  Southwest  Virignia Pocket-book  form.  2  00 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) Svo,  4  00 

Chester's  Catalogue  of  Minerals Svo,  paper,  i  00 

Cloth,  1  25 

Dictionary  of  the  Names  of  Minerals Svo,  3  50 

Dana's  System  of  Mineralogy Large  Svo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  Svo,  i  00 

Text-book  of  Mineralogy Svo,  4  00 

Minerals  and  How  to  Study  Them i2mo.  1  50 

Catalogue  of  American  Localities  of  Minerals Large  Svo,  i  00 

Manual  of  Mineralogy  and  Petrography i2mo,  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

Eakle's  Mineral  Tables Svo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small  Svo,  2  00 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses Svo,  4  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper,  50 
Rosenbusch's    Microscopical   Physiography    of   the    Rock-making  Minerals. 

(Iddings.) Svo,  5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo,  2  00 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia .» Svo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor.,  25  00 

Eissler's  Modern  High  Explosives Svo  4  00 

16 


3 

oo 

4 

00 

I 

50 

2 

50 

I 

oo 

3 

SO 

3 

00 

7 

50 

4 

00 

Fowler's  Sewage  Works  Analyses i2mo,  2  00 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  00 

**  lles's  Lead-smelting.     (Postage  gc.  additional.) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  treatment  of  Gold  Ores 8vo,  2  00 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

Hydraulic  and  Placer  Mining i2mo,  2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 


SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House i2mo, 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo, 

Water-supply  Engineering 8vo, 

Fuertes's  Water  and  Public  Health i2mo, 

Water-filtration  Works i2mo, 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo, 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo, 

Hazen's  Filtration  of  Public  Water-supplies 8vo, 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo, 

Examination  of  Water.     (Chemical  and  BacteriologicaL) i2mo,     i  25 

Ogden's  Sewer  Design i2mo,    2  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2nio,     i  25 

*  Price's  Handbook  on  Sanitation i2mo,    j  50 

Richards's  Cost  of  Food.     A  Study  in  bietaries i2mo,     i  00 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  00 

Richards  and  Woodman'-s  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  00 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 


MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanofif  and  Collins.).  .  .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Exctxrsion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound   i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894. .Small  8vo,  3  00 

Rostoski's.Serum  Diagnosis.     (Bolduan.) i2mo,  i  00 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  00 

17 


Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

Von  Behring's  Suppression  ot  Tuberculosis.     (Bolduan.) iimo,  i  00 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture:  Plans  for  Small  Hospital.  i2mo,  1  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  1  25 

Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible Sto,  2  25 

18 


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